\documentclass[12pt,reqno]{amsart} %\textwidth=6.5in \advance\oddsidemargin -.8700in %\advance\evensidemargin -.8700in \DeclareMathOperator{\ann}{ann}% \DeclareMathOperator{\can}{can}% \DeclareMathOperator{\chr}{char}% \DeclareMathOperator{\Gal}{Gal}% \DeclareMathOperator{\res}{res}% \DeclareMathOperator{\supp}{supp}% \DeclareMathOperator{\trdeg}{tr.deg}% \usepackage{amssymb,latexsym,amsmath,amscd} \usepackage[all]{xy} \begin{document} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \theoremstyle{definition} \newtheorem*{remark*}{Remark} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\Fp}{\mathbb{F}_p} \newcommand{\Jc}{\mathcal{J}} \newcommand{\Kf}{\mathfrak{K}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\mo}{\mathopen\langle} \newcommand{\mc}{\mathclose\rangle} \parskip=10pt plus 2pt minus 2pt \title[Galois Module Construction and Classification] {Construction and Classification of Some Galois Modules} \author[J\'{a}n Min\'{a}\v{c}]{J\'an Min\'a\v{c}$^{*\dagger}$} \address{Department of Mathematics, Middlesex College, \ University of Western Ontario, London, Ontario \ N6A 5B7 \ CANADA} \thanks{$^*$Research supported in part by the Natural Sciences and Engineering Research Council of Canada, and by the special Dean of Science Fund at the University of Western Ontario.} \thanks{$^\dag$Supported by the Mathematical Sciences Research Institute, Berkeley} \email{minac@uwo.ca} \author[John Swallow]{John Swallow$^\ddag$} \address{Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina \ 28035-7046 \ USA} \thanks{$^\ddag$Research supported in part by National Security Agency grant MDA904-02-1-0061.} \email{joswallow@davidson.edu} \begin{abstract} In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our description relies upon arithmetic invariants associated with $K/F$. Here we construct field extensions $K/F$ with prescribed arithmetic invariants, thus completing our classification of Galois modules $K^{\times}/K^{\times p}$. \end{abstract} \date\today \maketitle Let $F$ be a field of characteristic not $p$ containing a primitive $p$th root of unity $\xi_p$. For a cyclic field extension $K/F$ with Galois group $\Gal(K/F)$ of order $p$, let $J = {K^\times}/{K^{\times p}}$, and let $N$ denote the norm map from $K$ to $F$. In \cite{MS}, we proved that the structure of the $\Fp[\Gal(K/F)]$-module $J$ is determined by the following three arithmetic invariants: \begin{itemize} \item $d=d(K/F):=\dim_{\Fp} {F^\times}/{N(K^\times)}$, \item $e=e(K/F):=\dim_{\Fp} N(K^{\times})/F^{\times p}$, and \item $\Upsilon(K/F):=$ $1$ or $0$ according to whether $\xi_p\in N(K^\times)$ or not. \end{itemize} Now if $G=\Z/p\Z$, then $J$ may be considered an $\Fp[G]$-module via any isomorphism $G\cong \Gal(K/F)$, and the module structure of $J$ is independent of the choice of isomorphism. It is a fundamental problem to classify the isomorphism classes of modules $J$ for all $K/F$ in our context. This problem is solved in Theorem~\ref{th:mainth} below. Corollaries~1 and 2 of Theorem~\ref{th:mainth} describe all modules $J$ in an explicit way. In the following theorem we determine the sets of invariants $(d, e, \Upsilon)$ which may be realized by an extension $K/F$ and in so doing classify all $\Fp[G]$-modules $J=J(K/F)$ up to isomorphism. \begin{theorem}\label{th:mainth} Let $p$ be a prime number. For arbitrary cardinal numbers $d$, $e$, and for $\Upsilon\in \{0,1\}$, there exists a cyclic field extension $K/F$ of degree $p$ containing a primitive $p$th root of unity with invariants $(d, e, \Upsilon)$ if and only if \begin{itemize} \item if $\Upsilon=0$, then $1\le d$, \item if $p>2$ then $1\leq e$, and \item if $p=2$ and $\Upsilon=1$ then $1\leq e$. \end{itemize} \end{theorem} From the theorem above and from \cite[Theorem~3 and Corollary~2]{MS} we immediately obtain the following corollaries. We denote by $M_{i,j}$ the $j$th cyclic module $\Fp[G]$ such that $\dim_{\Fp}M_{i,j}=i$, where $j$ is a suitable index. \begin{corollary} \label{co:maincor1} Let $p>2$ be a prime number, and let $G$ be a cyclic group of order $p$. Then an $\Fp[G]$-module $J$ is realizable as an $\Fp[\Gal(K/F)]$-module $K^{\times}/K^{\times p}$ for some cyclic $G$-extension $K/F$ such that $F$ contains a primitive $p$th root of unity if and only if there exist cardinal numbers $d$, $e$, and $\Upsilon\in\{0,1\}$ such that \begin{itemize} \item[$(i)$] If $\Upsilon=0$, then $1\le d$; \item[$(ii)$] $1\leq e$; and \item[$(iii)$] \begin{equation*} J=\left(\bigoplus_{j\in\Kf_1}M_{1,j}\right) \bigoplus\left(\bigoplus_{j\in\Kf_2}M_{2,j}\right) \bigoplus\left(\bigoplus_{j\in\Kf_p} M_{p,j}\right), \end{equation*} where \medskip \begin{enumerate} \item[$(1)$] $\vert\Kf_1\vert+1=2\Upsilon+d$, \item[$(2)$] $\vert\Kf_2\vert=1-\Upsilon$, and \item[$(p)$] $\vert\Kf_p\vert+1=e$. \end{enumerate} \end{itemize} The invariants $d$, $e$, and $\Upsilon$ determine the module $J$ uniquely. \end{corollary} For $p=2$ using \cite[Theorem~3 and Corollary~3]{MS}, along with our theorem above, we obtain the next corollary. \begin{corollary} \label{co:maincor2} Now let $G$ be a cyclic group of order $2$. Then an $\F_2[G]$-module $J$ is realizable as an $\F_2[\Gal(K/F)]$-module $K^{\times}/K^{\times 2}$ for some quadratic extension $K/F$ with its arithmetic invariants $d(K/F)$, $e(K/F)$, and $\Upsilon(K/F)$ coinciding with $d$, $e$, and $\Upsilon\in\{0,1\}$, respectively, if and only if $d$, $e$, and $\Upsilon$ satisfy the conditions below. \begin{itemize} \item if $\Upsilon=0$, then $1\leq d$, \item if $\Upsilon=1$, then $1\leq e$. \end{itemize} In this case \begin{equation*} J=\left(\bigoplus_{j\in\Kf_1}M_{1,j}\right) \bigoplus \left(\bigoplus_{j\in\Kf_2}M_{2,j}\right) \end{equation*} where \begin{enumerate} \item[(1)] $\vert\Kf_1\vert+1=2\Upsilon+d$ and \item[(2)] $\vert\Kf_2\vert+\Upsilon=e$. \end{enumerate} Moreover, such a module $J$ is determined uniquely by the invariants $2\Upsilon+d$ and by $e-\Upsilon$ if $e$ is finite and by $e$ alone if $e$ is infinite. \end{corollary} If $p>2$ then two $\Fp[G]$-modules are isomorphic if and only if their invariants $d$, $e$, and $\Upsilon$ are the same, by \cite[Corollary~2]{MS}. Thus we see in particular that if $p>2$, then the arithmetic invariants of $J$ depend only upon the isomorphism type of the $\Fp[G]$-module $J$. In the case $p=2$, we see from \cite[Corollary~2]{MS} again that the arithmetic invariants $d$, $e$, and $\Upsilon$ determine our module $\F_2[G]$, but two isomorphic $\F_2[G]$-modules may have different arithmetic invariants; see \cite[Corollary~3]{MS}. Here is a very simple, concrete example illustrating this possibility. Let $K_1/F_1$ be a quadratic extension of finite fields of characteristic not $2$, $F_2=\R((t))$ be a field of power series with coefficients in real numbers $\R$, and $K_2=F_2(\sqrt{-1})$. Then both modules $K_{1}^{\times}/K_{1}^{\times 2}$ and $K_{2}^{\times}/K_{2}^{\times 2}$ are isomorphic to a trivial $\F_2[G]$-module $\F_2$, but their arithmetic invariants $(d_i, e_i, \Upsilon_i)$ are $(0, 1, 1)$ for $i=1$ and $(2, 0, 0)$ for $i=2$. \section{Notation and Strategy} \label{S1} In all that follows $F$ denotes a field, $F^{\times}= F\setminus\{0\}$ the multiplicative group of $F$, $p$ a prime number, and $F^{\times}/F^{\times p}$ the group of $p$th-power classes of $F$. For each $f\in F^{\times}$ we denote by $[f]$ the class of $f$ in $F^{\times}/F^{\times p}$. For each subset $A$ of $F^{\times}$ we denote by $[A]$ the set of classes $\{[a]\ \vert\ a\in A\}$ and by $\mo[A]\mc$ the subgroup of $F^{\times}/F^{\times p}$ generated by $[A]$. We denote by $\xi_p$ a primitive $p$th root of unity in $F$. (Some fields will be assumed to contain such a primitive $p$th root; for the other fields in this paper, we will prove that a primitive $p$th root is contained in the field.) Observe that our assumption that there exists a primitive $p$th root of unity implies that $\chr(F)\neq p$. For a Galois extension $K/F$, $\Gal(K/F)$ denotes the Galois group and $N_{K/F}$ denotes the norm map from $K$ to $F$. We denote by $F^s$ the separable closure of $F$ and $G_F$ the absolute Galois group $\Gal(F^s/F)$. As usual, $H^i(G_F, \Fp)$ are Galois cohomology groups of $F$ with coefficients in $\Fp$. Since all absolute Galois groups will be pro-$p$-groups, all considered $\Fp$ modules are trivial. Finally, let $\vert B \vert$ be the cardinal number of a set $B$. First observe that the conditions on $d$, $e$, and $\Upsilon$ listed in our theorem above are necessary: \begin{enumerate} \item If $\Upsilon=0$, then $\xi_p\notin N_{K/F}(K^{\times})$ and hence \begin{equation*} d =\dim_{\Fp} F^{\times}/N_{K/F}(K^{\times}) \geq 1; \end{equation*} \item If $p>2$ and $K=F(\root{p}\of{a})$ for a suitable $a\in F^{\times}\setminus F^{\times p}$, then $a= N_{K/F}(\root{p}\of{a})$ and hence \begin{equation*} e = \dim_{\Fp}N_{K/F}(K^{\times}) > 0. \end{equation*} \item If $p=2$, $\Upsilon=1$, and $K=F(\sqrt{a})$ for a suitable $a\in F^{\times}\setminus F^{\times 2}$, then $-1\in N_{K/F}(K^{\times})$ since $\Upsilon=1$. Consequently $a\in N_{K/F}(K^{\times})$, and thus $1\leq e$. \end{enumerate} Therefore in order to prove Theorem~\ref{th:mainth} when $p>2$, it is sufficient to show, for each cardinal numbers $d$, $e$ as above, for each $\Upsilon\in\{0,1\}$, and for each prime number $p>2$, the existence of a field $F$ such that: \begin{itemize} \item $F$ contains a primitive $p$th root $\xi_p$; \item $F^\times/F^{\times p}$ decomposes intro a direct sum of subgroups \begin{equation*} F^\times/F^{\times p} = D\oplus\mo[a]\mc\oplus E, \end{equation*} where $\dim_{\Fp}(\mo[a]\mc \oplus E) = e$ and, setting $K=F(\root{p}\of{a})$, \begin{enumerate} \item $[N_{K/F}(K^{\times})]=\mo[a]\mc \oplus E$; \item $\dim_{\Fp}(F^{\times}/N_{K/F}(K^{\times})) = \dim_{\Fp} D = d$; and \item $\Upsilon=0$ if and only if $\xi_p\notin N_{K/F}(K^{\times})$. \end{enumerate} \end{itemize} In the case $p=2$ and $\Upsilon=1$ we use the same conditions as above, and if $p=2$ and $\Upsilon=0$ we require instead that $e = \dim_{\F_2} E$ and $d=\dim_{\F_2}(D\oplus\mo[a]\mc)$. The latter condition is imposed because $-1\notin N_{F(\sqrt{a})/F} (F(\sqrt{a})^{\times})$ if and only if $a\notin N_{F(\sqrt{a})/F} (F(\sqrt{a})^{\times})$. Our strategy is to interpret the required conditions on $F^{\times}/ F^{\times p}$ above in terms of Galois cohomology. We then observe that these conditions are satisfied if $G_F$ is a free product, in the category of pro-$p$-groups, of suitable pro-$p$-groups $G_1$ and $G_2$, and finally we use the very nice theorem proved by Efrat and Haran which guarantees the existence of a field with $G_F$ above. This is one of the key results used in our paper. \begin{theorem} \label{th:efratharan} (Efrat-Haran; see \cite[Proposition~1.3]{EH}) Let $F_1,\dots,F_n$ be fields of equal characteristic such that $G_{F_1},\dots,G_{F_n}$ are pro-$p$-groups. Then there exists a field $F$ of the same characteristic such that \begin{equation*} G_F\cong G_{F_1}\star\cdots\star G_{F_n}, \end{equation*} where the product is free in the category of pro-$p$-groups. \end{theorem} \noindent In order to apply this theorem, we show the existence of the fields $F_1$ and $F_2$ such that $G_{F_1}$ and $G_{F_2}$ are prescribed Galois groups $G_1$ and $G_2$. We use the techniques of henselian valuations and formal power series to construct fields $F_1$ and $F_2$. \section{Lemmas} \subsection{Valued fields $F_1$ with prescribed residue field $F_0$ and valuation group $\Gamma$}\ Let $v$ be a valuation on a field $F_1$, written additively. Then we denote by $A_v$ the valuation ring $\{f\in F_1\ \vert\ v(f)\geq 0\}$; by $M_v$ the unique maximal ideal $\{f\in A_v\ \vert\ v(f)>0\}$ of $A_v$; by $F_v$ the residue field $A_v/M_v$ of $v$; by $\Gamma$ the valuation group $v(F_1^{\times})$ of $v$; and by $U$ the group $A_v\setminus M_v$ of units of $v$. The following lemma is well known and we shall omit its straightforward proof. \begin{lemma} \label{le:pthcount} Let $F_1$ be a valued field with valuation $v$, valuation group $\Gamma=v(F_1^\times)$, and group of units $U$. For each prime $p\neq \chr(F_1)$ there exists an isomorphism \begin{equation*} \varphi:F_1^{\times}/F_1^{\times p}\longrightarrow U/U^{p} \oplus \Gamma/p\Gamma. \end{equation*} In particular \begin{equation*} \dim_{\Fp} F_1^{\times}/F_1^{\times p} = \dim_{\Fp} U/U^{p} + \dim_{\Fp}\Gamma/p\Gamma. \end{equation*} \end{lemma} It is well-known that for each field $F_0$ and for each totally ordered abelian group $\Gamma$, there exists a field $F_1$ with a valuation $v:F_1\to\Gamma\cup\{\infty\}$ such that the residue field $F_v$ is isomorphic to $F_0$ and the valuation group is $\Gamma$. In order to construct such a field, set \begin{equation*} F_1 = F_0((\Gamma)) :=\{f:\Gamma\to F_0\ \vert\ \supp (f)\text{ is well-ordered}\}. \end{equation*} Thus a typical element $f\in F_1$ can be written as a formal sum $f=\sum_{g\in\Gamma} a_g t^g$ such that the set $\supp (f):=\{g\in G\ \vert\ a_g\neq 0\}$ is a well-ordered subset of $G$. The valuation $v$ on $f$ is defined as: $v(0)=\infty$ and $v(f)= \min \supp (f)$ for $f\neq 0$. An important property of the valued field $F_1$ as above is the fact that it is henselian. (See for example \cite[(1.3)]{Rib}.) In what follows we will identify $F_v$ with $F_0$. We will also assume that char $F_0\neq p$. We will be particularly interested in controlling the $p$th-power classes of such a field. To do so, we choose particular groups $\Gamma$ for our valuation groups. These groups will be direct sums of \begin{equation*} \Z_{(p)}:=\left\{\frac{a}{b}\in\Q\ \big\vert\ a,b\in\Z,b \neq 0; \ \text{if } a\neq 0 \text{ then } (a,b)=1, p\nmid b\right\}. \end{equation*} Observe that $\Z_{(p)}$ is the valuation ring of a $p$-adic valuation on $\Q$. Let $I$ be any non-empty, well-ordered set. Then set \begin{equation*} \Gamma=\Z_{(p)}^{(I)} := \left\{\gamma: I\to \Z_{(p)}\ \Big| \ \vert\supp(\gamma)\vert<\infty\right\}. \end{equation*} Thus $\Gamma$ is a direct sum of $\vert I\vert$ copies of $\Z_{(p)}$. Observe that $\Z_{(p)}$ carries a natural ordering induced from $\Q$, and then we may order $\Gamma$ lexicographically, as follows. Let $\gamma_1\neq\gamma_2\in\Gamma$. Then $\gamma_1<\gamma_2$ if and only if $\gamma_1(i)<\gamma_2(i)$ for the least element $i\in I$ such that $\gamma_1(i)\neq\gamma_2(i)$. Then $\Gamma$ is a linearly ordered abelian group. Recall that each non-empty set can be well-ordered. (See \cite[Appendix~2, Theorem~4.1]{La}.) We choose $\Gamma$ as above because $G_{F_0((\Gamma))}$ will be pro-$p$ (see Lemma~\ref{le:gf1prop} below) and because we may control the $p$th-power classes with the following lemma. This well-known lemma follows from Lemma~\ref{le:pthcount} and the fact that the valued field $F_1$ is henselian. It is also an immediate consequence of \cite[Lemma~1.4]{W}. Therefore we shall omit its proof. \begin{lemma}\label{le:pthcountf1} Let $F_1=F_0((\Gamma))$ as above. Then \begin{align*} \dim_{\Fp} F_1^\times/F_1^{\times p} &= \dim_{\Fp} F_0^\times/F_0^{\times p} + \dim_{\Fp} \Gamma/p\Gamma \\ &= \dim_{\Fp} F_0^\times/F_0^{\times p} + \vert I\vert. \end{align*} \end{lemma} Finally, we record a criterion for $G_{F_1}$ being pro-$p$: \begin{lemma} \label{le:gf1prop} Let $F_1=F_0((\Gamma))$ as above with $\chr(F_0) = 0$ and $G_{F_0}$ pro-$p$. Then $G_{F_1}$ is pro-$p$ as well. \end{lemma} \begin{proof} From basic valuation theory, nicely summarized in \cite[pages~3 and 4]{K}, and the fact that $F_1$ above is henselian, \begin{equation*} G_{F_1}\cong T\rtimes G_{F_0}, \end{equation*} where the action of $G_{F_0}$ on $T$ is uniquely determined by the cyclotomic character mapping $G_{F_0}$ into a group of automorphisms of a group of roots of unity contained in $F_0^s$, and $T\cong\Z_{p}^{I}$, the topological product of $\vert I\vert$ copies of $\Z_p$. In particular, if $G_{F_0}$ is a pro-$p$-group, so is $G_{F_1}$. \end{proof} \subsection{$H^2(G_{F_1}, \Fp)$ for henselian valued fields $F_1$}\ Now we study $H^2(G_{F_1}, \Fp)$ for our henselian valued fields $F_1$. The next lemma, taken from \cite{W}, will be used in the proof of Theorem~\ref{th:mainth} to show that norm groups of cyclic $p$-extensions of $F_1$ are not too large. Suppose that $F_1$ is a field endowed with a henselian valuation $v$ with valuation group $\Gamma=v(F_1^\times)$. Let $F_1^{{nr}}$ denote the maximal unramified extension of $F_1$ in its separable closure $F_1^s$. Then $G_{F_0}\cong G_{F_1}/\Gal(F_1^s/F_1^{{nr}})$. Therefore, after identifying these groups, we have the inflation map \begin{equation*} \inf = \inf{}_{F_0}^{F_1} : H^{*}(G_{F_0}, \Fp) \longrightarrow H^{*}(G_{F_1}, \Fp). \end{equation*} (See \cite[page~483]{W}.) Moreover, from basic Kummer theory we have the canonical isomorphism \begin{equation*} \varphi_F : F^{\times}/F^{\times p}\longrightarrow H^{1}(G_F, \Fp), \end{equation*} as well as the corresponding canonical isomorphisms $\varphi_{F_i}$, $i=1$, $2$. We will denote by $(f)_F$ or $(f_i)_{F_i}$ the images $\varphi_F([f])$ or $\varphi_{F_i}([f_i])$. If the context is clear we will omit the subscript. Assume next that $\{\pi_j,\ j\in \Jc\}$ is a set of elements of $F_1^{\times}$ such that their images in $\Gamma/p\Gamma$ form a basis of $\Gamma/p\Gamma$ over $\Fp$. Then we have the following lemma, obtained as a special case of a theorem of Wadsworth. \begin{lemma} \label{le:h2ofhens} \cite[Theorem~3.6, page~483]{W}. Let $F_1$ and $F_0$ be as above. Then \begin{align*} H^2(G_{F_1}, \Fp) = &\inf(H^2(G_{F_0},\Fp))\oplus_{j\in \Jc} \left( \inf(H^1(G_{F_0}, \Fp))\cup(\pi_j)\right) \\ &\oplus_{\{j_1,j_2\}\subset \Jc,\ j_1\neq j_2} \left((\pi_{j_1})\cup(\pi_{j_2})\right). \end{align*} Moreover, for each $j\in \Jc$, \begin{equation*} \inf(H^1(G_{F_0},\Fp)) \cong \inf(H^1(G_{F_0},\Fp)) \cup(\pi_j) \end{equation*} and for each $j_1,j_2\in \Jc$ such that $j_1\neq j_2$, we have $(\pi_{j_1})\cup(\pi_{j_2})\neq 0$. \end{lemma} Note that in the last summand of Lemma~\ref{le:h2ofhens} the sum ranges over subsets $\{j_1,j_2\},j_1\neq j_2$ of $\Jc$ and a choice between $(\pi_{j_{1}})\cup(\pi_{j_{2}})$ and $(\pi_{j_{2}})\cup(\pi_{j_{1}})$ is arbitrary but fixed. \subsection{Residue fields $F_0$ with prescribed absolute Galois group}\ In our construction of $F$ we choose a residue field $F_0$ depending on $\Upsilon$ and $p$. If $\Upsilon=1$ we will simply put $F_0=\C$, but when $\Upsilon=0$ we require some special properties of $F_0$. In particular, in order that our cyclic extension $K=F(\root{p}\of{a})$ have the desired invariant $\Upsilon=0$, we require that $K$ does not embed in a cyclic Galois extension $L$ over $F$ with degree $[L:F]=p^2$, for this is equivalent to $\xi_p\notin N_{K/F}(K^\times)$ by \cite[Theorem~3]{A}. To ensure that this nonembeddability condition holds, as well as to ensure that a certain nonabelian group of order $p^3$ does not occur as a Galois group over the field, we choose residue fields $F_0$ with absolute Galois groups taking a special form, and it is also convenient to require that $\vert F_0^\times/F_0^{\times p}\vert$ is small. As it turns out, we may choose some suitable algebraic infinite extension of $\Q$. Finitely generated pro-$p$-absolute Galois groups over $\Q$ and more generally any global field, were nicely classified in \cite{E2}. (See also \cite{E1} and \cite{JP} for related results and techniques.) The extensions we will need for $p>2$ are given in the following \begin{lemma} \label{le:specextsofQ} \cite[page~84]{E2} For each prime $p>2$ there exists an algebraic extension $F_{0,p}$ of $\Q$ such that \begin{equation*} G_{F_{0,p}} = \mo\sigma,\tau\ \vert\ \sigma\tau\sigma^{-1}= \tau^{p+1}\mc_{\text{pro-$p$}} \end{equation*} where the presentation is in the category of pro-$p$-groups. \end{lemma} Observe that the maximal abelian extension $F_{0,p}^{ab}$ of $F_{0,p}$ has $G_{F_{0,p}}^{ab} := \Gal(F_{0,p}^{ab}/ F_{0,p})$ equal to \begin{equation*}\label{eq:gfopab} G_{F_{0,p}}^{ab} \cong \mo\bar\sigma,\bar\tau\ \vert \ \bar\tau^{p}=1\mc=\Z_p\times\Z/p\Z \end{equation*} for all primes $p>2$. \subsection{Field arithmetic and free pro-$p$ products}\ In this section we collect lemmas giving information about a field $F$ derived from the structure of $G_F$, especially when $G_F$ is a free pro-$p$ product of two pro-$p$ groups $G_{F_1}$ and $G_{F_2}$. First we record a lemma detecting the presence of primitive $p$th roots of unity in a field $F$, based only on the structure of $G_F$. \begin{lemma} \label{le:pthrootexists} Suppose that $p>2$ and that $F$ is a field with $\chr(F)\neq p$ and $G_F$ pro-$p$. Then $\xi_p\in F^{\times}$. \end{lemma} \begin{proof} Because $\chr(F)\neq p$, there exists a primitive $p$th root $\xi_p$ of unity in $F^s$. If $\xi_p\in F^s\setminus F$ then $F(\xi_p)/F$ is a nontrivial Galois extension of degree $[F(\xi_p):F]>}[r]^{\can\ } & G_{F_1} \ar@{>>}[r] & \Z/p\Z. } \] Here the canonical map $\can$ is an identity on $G_{F_1}$ and contains $G_{F_2}$ in its kernel. Then the right-hand surjection and the composed surjection correspond to fields $K_1=F_1(\root{p}\of{a_1})$ and $K=F(\root{p}\of{a})$, respectively, where $(a)_{G_{F_1}}=(a_1)_{F_1}$ and $(a)_{G_{F_2}}=0$. \end{lemma} \begin{proof} The surjections are continuous homomorphisms, hence elements of $H^1(G_{F_1},\Fp)$ and $H^1(G_{F},\Fp)$, respectively, and the right-hand surjection is clearly the restriction of the composed surjection. The remainder follows by Kummer theory. \end{proof} \begin{lemma} \label{le:cupprod} Let $G_F=G_{F_1}\star G_{F_2}$ be pro-$p$ absolute Galois groups and suppose that $(a)_F$ and $(b)_F$ satisfy $(a)_{G_{F_1}} = (b)_{G_{F_2}} = 0$. Then \begin{equation*} (a)_F \cup (b)_F = 0 \in H^{2}(G_F, \Fp). \end{equation*} \end{lemma} The lemma follows from \cite[(4.1) Satz]{N} and from \cite[Prop.~7.3, page~191]{Ris}. However, we prove our lemma by translating the cup products into obstructions to basic Galois embedding problems, yielding an interesting Galois-theoretic variant of the proof. \begin{proof} If $(a)=0$ or $(b)=0$, we are done. Otherwise, the conditions $(a)_{G_{F_1}}=(b)_{G_{F_2}}=0$ imply that $(a)$ and $(b)$ are linearly independent in $H^1(G_F, \Fp)$. Now let $H_{p^3}$ be the Heisenberg group of order $p^3$: \begin{align*} H_{p^3} = \mo &v_1, v_2, w\ \vert\ v_1^p=v_2^p=w^p=1, v_2v_1=wv_1v_2, \\ &[v_1,w]=[v_2,w]=1 \mc \end{align*} In the case $p=2$, $H_{8}$ is the familiar dihedral group $D_4$. By \cite[Corollary, page 523 and Theorem 3(A)]{M}, if $(a)$ and $(b)$ are linearly independent, then $(a)\cup (b)=0$ if and only if $H_{p^3}$ is the Galois group $\Gal(M/F)$ of a Galois extension $M$ of $F$ containing $F(\root{p}\of{a}, \root{p}\of{b})$ in such a way that \begin{equation*} H_{p^3}/\mo v_1,w\mc = \Gal(F(\root{p}\of{a})/F) \text{ and } H_{p^3}/\mo v_2,w\mc = \Gal(F(\root{p}\of{b})/F). \end{equation*} Now consider the commutative diagram \[ \xymatrix{ & & G_F\ar@{>>}[ld]_{\delta_1} \ar@{.>>}[dd]^\beta \ar@{>>}[rd]^{\delta_2} & & \\ & G_{F_1} \ar@{>>}[dl]_{\alpha_1} & & G_{F_2} \ar@{>>}[dr]^{\alpha_2} & \\ \Z/p\Z \ar@{^{(}->}[rr]^{1\mapsto v_1} & & H_{p^3} \ar@(dr,dl)[rr]_{\mo v_1, w\mc\mapsto 0; \ v_2\mapsto 1} \ar@(dl,dr)[ll]^{\mo v_2, w\mc\mapsto 0; \ v_1\mapsto 1} & & \Z/p\Z \ar@{_{(}->}[ll]_{1\mapsto v_2} } \] Let $a_2\in F_2^\times$ and $b_1\in F_1^\times$ satisfy $(a_2)_{F_2} = (a)_{G_{F_2}}$ and $(b_1)_{F_1} = (b)_{G_{F_1}}$. Then set $K_1=F_1(\root{p}\of{b_1})$ and $K_2=F_2(\root{p}\of{a_2})$; these are $\Z/p\Z$-extensions of $F_1$ and $F_2$, respectively. We may then identify the left-hand $\Z/p\Z$ in the diagram with $\Gal(K_1/F_1)$ so that $\alpha_1$ is the surjection of Galois theory. Similarly, the right-hand $\Z/p\Z$ may be identified with $\Gal(K_2/F_2)$ with $\alpha_2$ the surjection of Galois theory. Finally, the topmost surjections $\delta_1$ and $\delta_2$ are canonical. By Lemma~\ref{le:resinterp}, the surjections $\delta_1\alpha_1$ and $\delta_2\alpha_2$ correspond to fields $F(\root{p}\of{b})$ and $F(\root{p}\of{a})$, respectively. Now because $G_F=G_{F_1}\star G_{F_2}$, there exists a homomorphism $\beta: G_F\to H_{p^3}$, and because $v_1$ and $v_2$ generate $H_{p^3}$, $\beta$ is a surjection. Hence $H_{p^3}$ is a Galois group over $F$ corresponding to a normal subgroup $H$ of $G_F$. Consider the smallest normal subgroup $H_1$ of $G_{F}$ containing $H$ and $G_{F_2}$. Then by the diagram, $G_F/H_1$ is the left-hand $\Z/p\Z$, which corresponds to $F(\root{p}\of{b})$, and $H_1/H$ is $\mo v_2, w\mc$. Now consider the smallest normal subgroup $H_2$ of $G_{F}$ containing $H$ and $G_{F_1}$. Then by the diagram, $G_F/H_2$ is the right-hand $\Z/p\Z$, which corresponds to $F(\root{p}\of{a})$, and $H_2/H$ is $\mo v_1, w\mc$. Hence $(a)\cup (b)=0$. \end{proof} Finally, we close with with a companion to Lemma~\ref{le:resinterp}. In Lemma~\ref{le:abeloffree} below, $\pi$ denotes the canonical homomorphism of $G$ to $G_1$ which is an identity on $G_1$, and is trivial on $G_2$. \begin{lemma} \label{le:abeloffree} Let $G=G_1\star G_2$ be a free product of $G_1$ and $G_2$ in the category of pro-$p$-groups. Suppose that $A\cong \Z/p\Z$ is a factor group of $G_1$ such that the surjection $G_1\to \Z/p\Z$ does not factor through $\Z/p^2\Z$. Then the following commutative diagram cannot occur: \[ \xymatrix{ G\ar@{>>}[r]^-{\pi} \ar@{>>}[d] & G_1 \ar@{>>}[d] \\ \Z/p^2\Z \ar@{>>}[r] & A. } \] \end{lemma} \begin{proof} Suppose that contrary to our statement, such a diagram as the above exists. Then by passing to quotients by commutator subgroups we obtain \[ \xymatrix{ G^{ab} \ar@{>>}[r]^-{\pi^{ab}} \ar@{>>}[d]_{\alpha} & G_1^{ab} \ar@{>>}[d]^{\gamma} \\ \Z/p^2\Z \ar@{>>}[r]^{\beta} & A. } \] But $G^{ab}\cong G_1^{ab}\times G_2^{ab}$ and the canonical surjection onto $G_1^{ab}$ is given by the projection map. Let $\delta$ be a splitting map of the projection map. Then $\gamma = \beta\alpha\delta$, contradicting the hypothesis. \end{proof} \section{Proof of the Theorem} \label{se:theproof} First we define fields $F_0$, $F_1$, $F_2$, and $F$ using our given cardinal numbers $d$, $e$, and $\Upsilon$, as well as the prime number $p$. Then we define the cyclic Galois extension $K/F$ of degree $p$ and check that the arithmetic invariants of $K/F$ coincide with $d$, $e$, and $\Upsilon$. \subsection{Constructing $F_0$, $F_1$, $F_2$, and $F$}\ If $\Upsilon=1$ then let $F_0=\C$. If $\Upsilon=0$ and $p=2$, let $F_0=\R$. (See Proposition~\ref{pr:trandeg} for alternative choices in these two cases.) If $\Upsilon=0$ and $p>2$ then let $F_{0,p}$ be the algebraic extension of $\Q$ of Lemma~\ref{le:specextsofQ}. In the first two cases we see trivially that $\xi_p\in F_0$, and in the last case $\xi_p\in F_0$ by Lemma~\ref{le:pthrootexists}. Observe that \begin{equation}\label{eq:f0pth} \dim_{\Fp} F_{0}^{\times}/F_{0}^{\times p}= \begin{cases} 0, \text{ if } \Upsilon=1;\\ 1, \text{ if } \Upsilon=0, \ p=2;\\ 2, \text{ if } \Upsilon=0, \ p>2. \end{cases} \end{equation} We next construct the field $F_1$. Because $\Upsilon=0$ implies $1\leq d$, for either choice of $\Upsilon\in \{0,1\}$ there exists a well-ordered set $I_1$ such that $\vert I_1 \vert+1=d+2\Upsilon$. Let $\Gamma_1=\Z_{(p)}^{(I_1)}$ be a direct sum of $\vert I_1 \vert$ copies of $\Z_{(p)}$. Then $\Gamma_1$ is a linearly ordered abelian group. Finally set $F_1 := F_0((\Gamma_1))$. From Lemmas~\ref{le:pthcountf1} and \ref{le:gf1prop} it follows that $G_{F_1}$ is a pro-$p$-group and \begin{equation*} \dim_{\Fp}F_{1}^{\times}/F_{1}^{\times p}= \dim_{\Fp}F_{0}^{\times}/F_{0}^{\times p} + \vert I_1 \vert. \end{equation*} Hence \begin{equation}\label{eq:f1pth} \dim_{\Fp} F_{1}^{\times}/F_{1}^{\times p}= \begin{cases} d+1, &\text{if } \Upsilon=1;\\ d, &\text{if } \Upsilon=0, \ p=2;\\ d+1, & \text{if } \Upsilon=0, \ p>2. \end{cases} \end{equation} Similarly, we construct $F_2$ as follows. Because $p>2$ and also $p=2$ and $\Upsilon=1$ implies $e>0$, there exists a well-ordered set $I_2$ such that $1+\vert I_2 \vert = e$ in either of the cases $p>2$ or $\Upsilon=1$, $p=2$, and $\vert I_2 \vert=e$ in the case $\Upsilon=0$, $p=2$. Then again $\Gamma_2=\Z_{(p)}^{(I_2)}$ is a linearly ordered abelian group. We set $F_2 := \C((\Gamma_2))$. Then from \cite[pages~3~and~4]{K} it follows that $G_{F_2}\cong\Z_{p}^{I_2}$, the topological product of $\vert I_2 \vert$ copies of $\Z_p$. In particular \begin{equation}\label{eq:edef} e = \begin{cases} \dim_{\Fp} F_{2}^{\times}/F_{2}^{\times p}+1, &\text{if } p>2 \text{ or } p=2 \text{ and } \Upsilon=1;\\ \dim_{\Fp} F_{2}^{\times}/F_{2}^{\times p}, & \text{if } p=2, \ \Upsilon=0.\\ \end{cases} \end{equation} From Theorem~\ref{th:efratharan} we see that there exists a field $F$ of characteristic zero, such that $G_F=G_{F_1}\star G_{F_2}$ is a free product of $G_{F_1}$ and $G_{F_2}$ in the category of pro-$p$-groups. In particular $G_F$ is again a pro-$p$-group, and from Lemma~\ref{le:pthrootexists} we see that $F$ contains a primitive $p$th root of unity. \subsection{Constructing $K/F$}\label{kf}\ We define the cyclic extension $K/F$ of degree $p$ as $K=F(\root{p}\of{a})$ where $[a]\in F^{\times}/F^{\times p}$ is chosen via the isomorphism \eqref{eq:res}. If $\Upsilon=1$ then let $(a_1)_{F_1}$ be any nontrivial element in $H^1(G_{F_1}, \Fp)$, which is possible since $1\leq \dim_{\Fp} H^1(G_{F_1}, \Fp) = d+1$. Let $(a_2)_{F_2}=0$ in $H^2(G_{F_2}, \Fp)$. Now suppose $\Upsilon=0$ and $p>2$. We denote the fixed field of the factor $\Z_p$ in $G_{F_0}^{ab}\cong\Z_p\times\Z/p\Z$ acting on a maximal abelian extension $F_{0}^{ab}$ of $F_0$ as $K_0:=F_0(\root{p}\of{b})$. (See the discussion following Lemma~\ref{le:specextsofQ}.) Because $F_{0}^{\times}/F_{0}^{\times p}$ is naturally isomorphic to a subgroup of $F_{1}^{\times}/F_{1}^{\times p}$, we may set $(a_1)_{F_1} = \inf (b)_{F_0}\neq 0$ in $H^1(G_{F_1}, \Fp)$ and $(a_2)_{F_2}=0$ in $H^2(G_{F_2},\Fp)$. Finally assume that $\Upsilon=0$ and $p=2$. We set $(a_1)_{F_1}=(-1)_{F_1}\neq 0$ in $H^1(G_{F_1}, \F_2)$ and $(a_2)_{F_2}=0$ in $H^2(G_{F_2},\F_2)$. Now by \eqref{eq:res} there exists $a\in F^\times$ such that $(a)_F\neq 0$ and $(a)_{G_{F_i}}=(a_i)_{F_i}$, $i=1$, $2$. Since we have already determined that $\xi_p\in F^{\times}$, we see that $K=F(\root{p}\of{a})$ is a cyclic extension of degree $p$. Hence it remains to show that the arithmetic invariants $d(K/F)$, $e(K/F)$, and $\Upsilon(K/F)$ coincide with the prescribed cardinal numbers $d$, $e$, and $\Upsilon$ respectively. \subsection{Determining $d(K/F)$ and $e(K/F)$ via annihilators}\ We first observe some relationships among cup products of elements in $H^1(G_F, \Fp)$. Recall that Lemma~\ref{le:cupprod} tells us that if $c_1, c_2\in F^\times$ with $(c_1)_{G_{F_1}} = (c_2)_{G_{F_2}} = 0$ then \begin{equation}\label{eq:f1f2} (c_1)_F \cup (c_2)_F = 0\in H^2(G_F, \Fp). \end{equation} Moreover, recall that by \cite[Satz~4.1]{N} \begin{equation*} H^2(G_F,\Fp)\cong H^2(G_{F_1},\Fp)\times H^2(G_{F_2},\Fp), \end{equation*} where the isomorphism is induced by the restriction maps \begin{equation*} \res_i : H^2(G_F,\Fp) \to H^2(G_{F_i}, \Fp), i=1, 2. \end{equation*} Because these restriction maps commute with cup product maps (\cite[Proposition~7.3, page~191]{Ris}), we see that if $c_1, c_2\in F^\times$ such that $(c_1)_{G_{F_2}}=(c_2)_{G_{F_2}}=0$ in $H^2(G_{F_2}, \Fp)$ then \begin{equation}\label{eq:tof1} (c_1)_F \cup (c_2)_F \neq 0\quad \text{iff}\quad (c_1)_{G_{F_1}}\cup (c_2)_{G_{F_1}}\neq 0, \end{equation} where the first cup product lies in $H^1(G_F, \Fp)$ and the second cup product lies in $H^1(G_{F_1}, \Fp)$. By symmetry the analogous statement with $F_1$ and $F_2$ exchanged holds as well. Now we calculate $e(K/F)$. We adopt the following notation for an annihilator of $\phi\in H^1(G_{F},\Fp)$: \begin{equation*} \ann_{F_1}\phi := \{\eta\in H^1(G_{F}, \Fp)\ \vert\ \eta\cup\phi=0\}. \end{equation*} For $x\in F^{\times}$ we have $x\in N_{K/F}(K^{\times})$ if and only if $(x)_F \cup (a)_F =0$ in $H^2(G_F, \Fp)$. Therefore, using \eqref{eq:f1f2} and \eqref{eq:tof1} with the fact that in every case $(a)_{G_{F_2}}=0$, \begin{align*} e(K/F) &= \dim_{\Fp} N(K^{\times})/F^{\times p} \\ &= \dim_{\Fp}\{(x)_F \in H^1(G_F,\Fp)\ \vert \ (a)_F \cup (x)_F=0\} \\ &= \dim_{\Fp}\{(y)_{F_1} \in H^1(G_{F_1}, \Fp)\ \vert \ (a)_{G_{F_1}} \cup(y)_{F_1}=0\}\\ &\phantom{=\ \ } + \dim_{\Fp} H^1(G_{F_2},\Fp) \\ &= \dim_{\Fp} \ann_{F_1}(a)_{G_{F_1}} + \dim_{\Fp} F_2^\times/F_2^{\times p}. \end{align*} If $\Upsilon=1$ then $H^1(G_{F_0}, \Fp)=0$, and so by Lemma~\ref{le:h2ofhens} we have $\dim_{\Fp} \ann_{F_1} (a)_{G_{F_1}} = 1$. If $p>2$ and $\Upsilon=0$ then by Lemma~\ref{le:h2ofhens}, \begin{equation*} \dim_{\Fp} \ann_{F_1}(a)_{G_{F_1}} = \dim_{\Fp} \ann_{F_0} (b)_{F_0}, \end{equation*} where $b$ was chosen so that $K_0=F_0(\root{p}\of{b})$ was the fixed field of the factor $\Z_p$ in $G_{F_0}^{ab}$. Now because $(b)_{F_0} \cup (b)_{F_0} = 0$ as an identity in $H^2(G_F, \Fp)$ for $p>2$, \begin{equation*} 1\le \dim_{\Fp} \ann_{F_0} (b)_{F_0} \le 2. \end{equation*} On the other hand, let $c\in F^\times\setminus F^{\times p}$ be defined so that $F_0(\root{p}\of{c})$ is contained in the fixed field of the factor $\Z/p\Z$ in $G_{F_0}^{ab}$. Then by \eqref{eq:f0pth}, $\{ (b)_{F_0}, (c)_{F_0}\}$ spans $H^1(G_{F_0}, \Fp)$. Furthermore, since by Lemma~\ref{le:specextsofQ}, $H_{p^3}$ is not a quotient of $G_{F_{0,p}}$, $(b)_{F_0} \cup (c)_{F_0}\neq 0$ by \cite[Corollary, page 523 and Theorem 3(A)]{M}. We conclude that $\dim_{\Fp} \ann_{F_0} (b)_{F_0} = 1$. Finally, if $p=2$ and $\Upsilon=0$ then again by Lemma~\ref{le:h2ofhens} \begin{equation*} \dim_{\F_2} \ann_{F_1}(a)_{G_{F_1}} = \dim_{\F_2} \ann_{F_0} (-1)_{F_0}. \end{equation*} As $F_0=\R$, $\dim_{\F_2} F_0^\times/F_0^{\times 2} = 1$ and $(-1)_{\R}\cup (-1)_{\R} \neq 0$, yielding $\dim_{\F_2} \ann_{F_1} (a)_{G_{F_1}} = 0.$ Combining our results with \eqref{eq:edef}, we have that $e(K/F)=e$. Now we turn to a similar calculation of $d(K/F)$. Again using \eqref{eq:f1f2} and \eqref{eq:tof1} with the fact that in every case $(a)_{G_{F_2}}=0$, \begin{align*} d(K/F) &= \dim_{\Fp}(H^1(G_F,\Fp)/ \ann_{F} (a)_F) \\ &= \dim_{\Fp}(H^1(G_{F_1},\Fp)/\ann_{F_1} (a)_{G_{F_1}}) \\ &= d. \end{align*} For this last equality we use \eqref{eq:f1pth} together with calculations of the dimension of $\ann_{F_1} (a)_{G_{F_1}}$ already achieved. Hence $d(K/F)=d$ in all cases. \subsection{Determining $\Upsilon$ via quotients of $G_F$}\ It remains to show that $\Upsilon(K/F)=\Upsilon$. First consider the case $\Upsilon=1$. Since $F_0=\C$, $\xi_p$ is a $p$th power in $F_1$, and $F_1(\root{p}\of{a})$ embeds in a $\Z/p^2\Z$-extension $F_1(\root{p^2}\of{a})$ of $F_1$. Then the surjection $G_{F_1}\to\Gal(K_1/F_1)$ factors through $\Z/p^2\Z$. Following the surjection with the canonical surjection $G_F\to G_{F_1}$, we see that $\Z/p^2\Z$ is a factor group of $G_F$. Moreover, by Lemma~\ref{le:resinterp}, the surjection $G_F \to \Z/p\Z$ corresponds to $K$. Hence $K/F$ embeds in a $\Z/p^2\Z$-extension of $F$. By \cite[Theorem~3]{A}, $\xi_p\in N_{K/F}(K^{\times})$. Therefore $\Upsilon(K/F)=1=\Upsilon$. Now consider the case $\Upsilon=0$ and $p>2$. Because $K_0=F_0(\root{p}\of{b})$ does not embed in a $\Z/p^2\Z$-extension of $F_0$, $\xi_p\notin N_{K_0/F_0}(K_0^\times)$. (See \ref{kf} for the definition of $K_0$.) Hence $(b)_{F_0} \cup(\xi_p)_{F_0}\neq 0$ in $H^1(G_{F_0},\Fp)$, and, by Lemma~\ref{le:h2ofhens}, $(\inf (b))\cup (\xi_p)_{F_1} \neq 0$ in $H^1(G_{F_1},\Fp)$ as well. Choose $b_1\in F_1^\times$ so that $(b_1)_{F_1} = \inf (b)_{F_0}$ and set $K_1=F_1(\root{p}\of{b_1})$. Then $\xi_p \notin N_{K_1/F_1} (K_1^\times)$ and by \cite[Theorem~3]{A} the field extension $K_1/F_1$ does not embed in a $\Z/p^2\Z$-extension of $F_1$. Therefore the surjection $G_{F_1}\to \Gal(F_1(\root{p}\of{b})/F_1)$ does not factor through $\Z/p^2\Z$. From Lemmas~\ref{le:resinterp} and \ref{le:abeloffree} we see that surjection $G_F\to\Gal(K/F)$ does not factor through $\Z/p^2\Z$. Again by \cite[Theorem~3]{A} we conclude that $\xi_p\notin N_{K/F}(K^{\times})$ and therefore $\Upsilon(K/F)=0=\Upsilon$. Finally consider the case $\Upsilon=0$ and $p=2$. Because $-1\notin N_{\C/\R}(\C)$ from Lemma~\ref{le:h2ofhens} we conclude that $-1\notin N_{K_1/F_1}(K_{1}^{\times})$. By \cite[Theorem~3]{A} the surjection $G_{F_1}\to\Gal(K_1/F_1)$ does not factor through $\Z/4\Z$. As in the previous case, by Lemmas~\ref{le:resinterp} and \ref{le:abeloffree} we see that $K/F$ does not embed in a $\Z/4\Z$-extension of $F$. Again by \cite[Theorem~3]{A} we see that $-1\notin N_{K/F}(K^{\times})$ and therefore $\Upsilon(K/F)=0=\Upsilon$. Hence we have checked in all cases that $\Upsilon(K/F)=\Upsilon$, and our proof of Theorem~\ref{th:mainth} is now complete. \qed \begin{remark*} In \cite[Lemma~1.2 and Proposition~1.3]{EH} Efrat and Haran construct some fields with prescribed absolute Galois groups together with some bounds on the transcendence degrees of these fields. These bounds, together with the replacement of $\C$ by an algebraic closure of $\Q$ and of $\R$ by a real-closed algebraic number field $R$ in the case when $p=2$ and $\Upsilon=0$ in our proof above, yield the following proposition. \end{remark*} \begin{proposition} \label{pr:trandeg} Suppose that $p$ is a prime number, and let $d$, $e$, and $\Upsilon\in\{0,1\}$ be cardinal numbers such that if $\Upsilon=0$ then $1\leq d$, if $p>2$ then $1\leq e$, and if $p=2$ and $\Upsilon=1$ then $1\leq e$. Then there exists a field $F$ containing $\Q(\xi_p)$ and a cyclic Galois extension $K$ of degree $p$ over $F$ such that \begin{equation*} e(K/F)=e, \quad d(K/F)=d, \quad \text{and }\quad \Upsilon(K/F) = \Upsilon. \end{equation*} Moreover \begin{equation*} \trdeg (F/\Q)\leq 1+ \max\{e,d+1\}. \end{equation*} In particular if $d, e\in\N\cup\{0\}$, there exists a Galois cyclic extension $K/F$ of degree $p$ with prescribed invariants $d$, $e$, and $\Upsilon$ of finite transcendence degree over its prime field $\Q$. \end{proposition} \section{Acknowledgements} \label{S4} We would like to thank the organizers of the MSRI programs on Galois theory and the MSRI staff for giving us the opportunity to meet and to begin our collaboration in the Fall of 1999. We are also grateful to A.~Wadsworth for stimulating conversations. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /notbar put dup 67 /C put dup 70 /F put dup 78 /N put dup 81 /Q put dup 82 /R put dup 90 /Z put dup 111 /multicloseright put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: XYBSQL10 %!PS-AdobeFont-1.1: XYBSQL10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 17 22:48:36 % % XYBSQL10: quarter circle segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly= def /FullName (XYBSQL10) readonly def /FamilyName (XYBSQL) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBSQL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 31 /d31 put dup 63 /d63 put dup 95 /d95 put dup 127 /d127 put readonly def /FontBBox{-376 -376 376 376}readonly def /UniqueID 5092842 def currentdict end currentfile= eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A529226 454434A66FB600391E72DBCB27066ACC4F037CF32BDBE1D3FEDB2F47ED471216 614265A890F90F735C48E50F819CB3328DEF7288798EC4F2C110FC584615DA50 CB3270F8E3CFDCF89F6D8F4A766AE1BE4D71002FDE064A5DB86C717534590905 8FB00A60B579BEB0FBC24E561A610384F50BA13268295F448B37D49AE5E5897F FD7C4D05E340C9755AAB4B78EA191EB05FB2034AEB8CD021AB4B51CD01485AD2 8AA14D2B090A404B50F59937CFC161701277096CCD53099455F93A771007C77B B87783C77E9E2F69FF3A048B67BC906F620FE1560DF1A83A14B79CD2113826A3 843E73B57B31141B021CA4EF1773AA255B72EF7C6CBDE338904C0826636267AB D903468094834C1A6AA0D10C380761F91CC6116D3734356E3E265D7310842F2F 56A99E24ACD9B4B1779C020BA15D67425303B715B1A988B5931BB9D635766EC3 570B31E14384959712DADBF01F9F3982BEFB27370950B8FC6D30E18570C7D8CD 4D5CC9C7893D2CDFF0F7CDB204303651FF06EA526B6C047501AB814404AECFD0 CD1957CA857C76112108E0BA654303DA70EDCCDEFFCD4FEEC5D0A7FB703F33D9 FD4DB64E2E47C2FDA7F6A46D2DB43E8A23DF3AA9151C8A3B5EBA45C19BC5C3F6 F96B20F83E6A0C10E0F50AA332B3C9ECAEB132C2973D8B5AF7FDB038087E4025 7C1FE6E3C5FC633D04DBBFE79838ABD96393713957F4ACA332BDC8DB35A66E89 08198D98C2347F88FD9C24A003C23DE44D0A1218E6B32D8D6B1BC589266B4332 DB4CB01B5B920CAF661D96F571917A6D073AFC1210110325767593E7C3FB4F04 0DB5BFC009ADDB6B3CC5ED11E892077C25199F3875A0D68EF086D1072AB0649C B0C032A36AD71F04909EED1C616389918B24A4E6C054374758BB635966442A41 19C75F14E864B1EA48DCC836A8E1F2BEFFE9DFA7E417D4A9F608AC6D7B35AFD6 F5B13948F568EE579E7F59B5FEE97F7763900A1DFB51B1B8A61CF94FA0C3F624 4750943626CA75F8A9769D13892C42E4A98EA242AB3992AAE9BA3CEBDA784C09 4D009323FF8E612EC3433E14BE80BDBC7143CC0DE847D85943C5DDB74FBCF1D1 2B612030BA88A3FCDEEC6D284526B5092B973A220129C4435CD76B4837D5D8EA C09BE4E529E83CFB266C5DA7063352B0595C1F01D2632BF7765218F08F36208F 257AB42E3574C95360CE4974E88B89E4AEC1687558C61DF0550EE352A7B6D443 EBE459079DB4BED9C9BF8ADADBBFE8B74C8315639814D00260C0D89134ECA76E 81FB23DD02DDAD17F94EB4DD7D0DD8783330B9AB565E49B1DA05BE359831310E 47874BF67F980F48B895310DA844C70BA8AAEEF9F19C6C60AB17E09DC224792A 51EE5FFC9DAA83BA7EB97A3A5F66E7F07C9C389579E6E07C7B8517CFC535DB0D 12B6F57A7C292582309C09E89CF4111734BAA78CAB3BDD6B34C23235968C1485 1E3760940A38F367A42D48D453 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYDASH10 %!PS-AdobeFont-1.1: XYDASH10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Aug 28 05:34:12 %%RevisionDate: 1997 Sep 18 10:23:31 % % XYDASH10: line segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly= def /FullName (XYDASH10) readonly def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -300 def /UnderlineThickness 150 def end readonly def /FontName /XYDASH10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 68 /d68 put dup 70 /d70 put dup 120 /d120 put dup 122 /d122 put readonly def /FontBBox{-40 -520 503 520}readonly def /UniqueID 5092844 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: XYBTIP10 %!PS-AdobeFont-1.1: XYBTIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly= def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType=200 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /d15 put dup 34 /d34 put dup 35 /d35 put dup 47 /d47 put dup 56 /d56 put dup 102 /d102 put dup 111 /d111 put dup 123 /d123 put dup 124 /d124 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueID 5092839 def currentdict end currentfile= eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F409B932A74D078F654BDC704DE35B7CC455A4DC3 FED12AD5A138F12D1F36AB94792D7CBBABAB576094BB3DACD5249F299F6A44E5 D55493440B9008DAD1724D1025FF4C1B847BE604D73EB4978213D57EEAD8A8D1 5A86CA35DD6601510434BA8FFED4C13D902896B29BB9785C3082D736BA9823D5 7E64CA23FEA7F44B2124E5F67CEC97E9DE58B6FA5B980B36AE286B081C98D7BD 9879B11A89E194E2DC14674A18FDD0A8E717A0A56E3FC56D5D6D655F4A9249DB 156F3F9E90103C2B60AC987D0264913C3968879B4C420AD75F215263391DA687 FF58D5E85224C89D255EE2D1CE35F12658D97B5447970E044DECC9770B3E1D15 08A427888A7583890516C231E025873CB1971633D6A13245461D4F18E3169505 5D4C5AEFEDE1C6F5F1CE254FB5302704D8ED60058E645FF630B23D25D6C70C44 330963D29CA7058CCD18114B4EF9C9AB72AD444484D7CBCD2EE82742C558B477 0182F1B77892DF07F2189550DFD355042347E288EE60AAC93A4A8F364DCD7582 63569E845E521F361A8E0AE4CDB4A73DE296E8D0D99CBC0E2407F01C465D6201 5492CBC3309FF112E164B5182ACD1BCC7616E1577A23FA086C7191A986041E13 967F12567DA35A9F58F59C298C9D8BB023B1AC9F705CA45B31A132C3BD2BF793 71B1A4373E71606C132C05A88474875C3E3E963660814FC12497D7A39ABAD71E 6F08A07F236B72277178119C9011DBE677E6F550F9392455C07518C027D23C5C 1344CE300AAF4404A774D4A1F7768E8EEB045857C99020FB4524C1E566C2A397 2A9F6E0385C8B51517D0A1A919A5B54FCB1B3DE4DA78EA5DAF403363900C2BC4 22F2E3F39A44BB67EA0DE171EC0B40C549FB6A54A3A4E0676F50469391F07C3E 1D1C265EB77E07208CC9904011E7582FFA75EFEA014883B4E112BC677BFEB9EC E0CE21D51E69157A9EE90C1F707DC35B2B53208A06B3BDD394C7A7B52F551F5D FD7185742A4873B04B503F44E0AE8ED7D5E53A1A0F4EB0FEEC49DE3944479AB4 D0CEA8C94E230E14EEFD8DEC36247B638A0849CBC1382CAD72E1447EF0332546 813B27521C685ADC9F47BEFE3720747044E42D613DCE22C81C797A811F10704C 8BCA87720F77E3F12FB449408E837E5D32451C9C40DF6C54C02BC0F134D42CC2 1868216B2EE56FD10D031A41532B4C2F3D5486125212396FC52741FE404C158A D42D84644DDA14134A90BCFD790425195376D2CB47E86999D80E14F881E9E942 F5D0E0D3133CBAEAF3E8881A482BBE09B2DD382F30188981F41CB10B8D9EE99C AC04BFFB03D0DDCB54BD53D7BB0D106B4597C883EE4FA787961C069F0F4E89C5 F2AC332D7DF7E0D86AFDA22AFFBA34336BC918A1F4E076FE338329CBE5F26775 0E727BED856DBBDBDD228A54BA126A5671304391E8161F1372B0EB310A43664F EAAB069BD22E80A72290A828C30CCD96ED016BA23895269B085F92C691F1FBF1 F6F78D2E838B1CE1FF5C6CDF8FE3BBBB5641A4F4DF8264DC1A2C2446789DB4D5 BE7F7B9C823FEB0C1C0D34075266CCCEAD126CC58D1006572A8E0F740520123E 220A13E990B8F688B45529D6B54C86C302323BAB5800F10B984C76FF6C79DD52 5164303317474FCBC25176C02C24BA0F038D26DCEA2058D8ADA4D5A3767116B1 29B0936F72A57FA1E7AA1FFE65DC930416590A89DF3987446F31241DD91690B9 766DF54E8C0CB3792C351BBDA85793FE72FE510843A778F5AC109241AAC2582A A23F2EFBCA86AA39973DBCE01CF21512CAF09028F43C43E2FEB7F41B92E12B20 C0B7A2F80BC1E429577AD0613D5CD90EBD11AD8828D9256344DA47964F72ADBA 3BE76089428B831438739B2F23BC205E072393821BE1D6AEB1B9EB2410D7F70A B6B13B7D160131545475D5DF2911AA051AFFD92163BC6C93634B515145F58C28 10FFAB11EEAE2155DDCA68342F7B4CD32474C01530C2E38A121F7ED5E48396C8 AF8029773468DE2522E0084366CA56038814D8801AFF1979D59CA330FBA69A68 940503D894B0479D86F4C87ABF727F61D8724A94F06DDC5FDB6BF178608F95A9 05F3272F4BB514FB8DC0E84538622F19DF2A0F0B78029B69A0DD6703B33CC5D6 6AC681EEFED78F3D36626170F4D60EA68222A2C8CF187815F292116D574865D5 15EAD211F38473156981DD190BF4973D871C2147709EE2438C4E575C900A03BC 071DA5544B37595EBAD512451067176981A96168EE13EF0CC1BF6633D3E8D8C4 888FA2A7AA449B5AB439E0F7B7C790707273C291B204C19F28C44C1389955D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly= def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYATIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /d15 put dup 34 /d34 put dup 35 /d35 put dup 47 /d47 put dup 56 /d56 put dup 102 /d102 put dup 111 /d111 put dup 123 /d123 put dup 124 /d124 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueID 5092838 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 111 /o put dup 112 /p put dup 114 /r put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A30EB76029337 900ECFB1390CA5C0C3A04528044F266BA17BE487C79B94FAC6D6484684C5BFEA 87BCCC77D40AD11552035E95E3007126418ED49B68468B38A14E88E68A267B98 076F1C9769A5AFBC285E5B158EAC9F926F1D6C0B8F1D57D9C31D25AE27123518 9D2CD92E5689E0213089BD268DA5E47525CB8EABAA4B78A15AEA34705889AB3A FFB8953B5B3482E52BFA0940630ADF8C0AC2177D907324299EE980E850F203CD B627962F43D5A678C44243CDE97853BDC6AB45FD5C09AD274DAF89929F583CC9 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 75 /K put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 56 /eight put dup 59 /semicolon put dup 61 /equal put dup 97 /a put dup 99 /c put dup 110 /n put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 75 /K put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueID 5031992 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 59 /semicolon put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 20 /caron put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 70 /F put dup 97 /a put dup 98 /b put dup 105 /i put dup 110 /n put dup 112 /p put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 70 /F put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 8 /circleplus put dup 15 /bullet put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 74 /J put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 26 /propersubset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 74 /J put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 25 /pi put dup 59 /comma put dup 61 /slash put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 97 /a put dup 98 /b put dup 103 /g put dup 105 /i put dup 106 /j put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 118 /v put dup 119 /w put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put 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57CB65FE968A3B6FA2E6A4A85F4026AB20F02CED64D70DDF50EE21F2E5B18720 0A1D1F622B5C92243EB30A9239B63D4DFC472D9AC21F96CA339D8A325CE18B81 3B0FCD2CC6A803CD3A965C371CFD0B64186CCF8C9E40D506B42B54EEC8D21E4E 2B8FF2C7542FDB59405949F1711F393E9C81B517F5469DC9D19C716CCB268BD9 03DB33DCAAE5A370BC1C9DC729D9AE4F5F74F665023DE2A1595D32E8A9C2BB0B DC038BE4A62E673F17BEAF79FC308801A52032CD23D6375F657E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding=20256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 7 /Upsilon put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 112 /p put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 61 /slash put dup 70 /F put dup 75 /K put dup 112 /p put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile= eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /multiply put dup 3 /asteriskmath put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights= Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 19 /acute put dup 20 /caron put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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1[35 35[55 1[20 2[35 3[35 35 35 35 35 4[55 1[27 27 39[44{}15 66.4176 /CMR8 rf /Fm 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Fn 180[46 75[{}1 66.4176 /EUFM7 rf /Fo 144[66 66 29[105 2[151 11[87 87 1[89 1[89 1[89 1[89 6[87 87 7[80 6[79 79 19[33 10[46 46{}18 99.6264 /CMEX10 rf /Fp 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 12[70 55 2[66 1[72 87 61 2[38 72 75 64 66 9[30 30 11[30 35 30 2[40 40 27[55 12[{}45 99.6264 /CMTI12 rf /Fq 168[79 1[58 55 43 57 2[60 58 70 48 1[39 28 58 60 50 52 59 55 1[58 7[38 38 38 38 38 38 38 38 38 38 27[38 38 19[{}30 74.7198 /CMR9 rf /Fr 133[34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 1[25 42 38 25 38 42 38 38 42 9[83 3[47 61 1[56 64 2[52 64 1[32 62 64 54 56 63 59 1[62 6[25 11[25 30 25 4[25 19[42 6[47 51 11[{}48 83.022 /CMTI10 rf /Fs 143[32 1[38 4[22 6[27 34 26[39 10[19 59[{}7 49.8132 /CMMI6 rf /Ft 173[48 11[41 2[48 67[{}3 66.4176 /MSBM7 rf /Fu 143[83 1[50 3[28 39 39 50 50 10[66 16[68 19[0 3[66 100 15[100 8[77 2[77 77 4[50 6[77 5[77 28 77{}21 99.6264 /CMSY10 rf /Fv 144[77 20[66 7[72 77 2[72 7[61 2[72 21[28 45[{}8 99.6264 /MSBM10 rf /Fw 143[59 6[27 27 35 35 27[48 18[0 0 3[47 16[71 6[55 22[35 55 1[55{}14 66.4176 /CMSY8 rf /Fx 136[51 34 2[33 32 1[36 1[43 3[29 24 1[34 4[30 37 21[60 1[31 1[55 45 8[35 1[20 33[41 10[31 36 40 45 11[{}22 66.4176 /CMMI8 rf /Fy 134[48 55 70 47 1[35 3[49 5[40 33 1[47 48 45 51 42 41 51 11[66 57 1[74 3[78 94 66 83 54 43 81 77 63 72 81 1[74 73 1[49 76 49 76 27 27 18[64 8[58 1[42 55 1[55 43 6[48 2[43 51 55 62 11[{}48 99.6264 /CMMI12 rf /Fz 133[43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 3[27 1[27 3[100 2[70 54 72 1[66 76 73 89 61 76 50 35 73 77 64 66 1[70 69 73 3[76 1[27 27 49 49 49 49 49 49 49 49 49 49 1[27 33 27 76 1[38 38 17[49 7[81 54 54 57 3[76 6[61{}73 99.6264 /CMR12 rf /FA 143[34 112[{}1 58.1154 /CMMI7 rf /FB 143[42 36[71 4[53 8[42 61[{}4 83.022 /CMMI10 rf /FC 134[51 51 1[51 1[49 38 50 1[46 53 51 62 43 2[25 51 53 44 46 52 49 48 51 9[92 1[68 1[51 3[70 68 82 57 7[69 65 64 68 8[46 46 46 46 46 46 46 1[46 1[27 31 27 44[{}42 83.022 /CMCSC10 rf /FD 133[30 30 117[34 52 2[{}4 58.1154 /CMSY7 rf /FE 132[42 1[44 44 60 44 46 32 33 33 1[46 42 46 69 23 44 1[23 46 42 25 37 46 37 46 42 3[23 1[23 1[62 1[85 62 62 60 46 61 65 57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 6[23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 2[32 32 23 18[42 42 6[46 12[{}70 83.022 /CMR10 rf /FF 134[59 1[81 59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 10[85 86 78 62 84 1[77 84 88 106 67 2[42 1[88 70 74 86 81 1[85 7[56 56 56 56 56 56 56 56 56 2[31 37 31 31[62 12[{}53 99.6264 /CMBX12 rf= end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 538 716 a FF(CONSTR)m(UCTION)47 b(AND)h(CLASSIFICA)-9 b(TION)48 b(OF)h(SOME)1429 832 y(GALOIS)g(MODULES)1186 1081 y FE(J)1239 1060 y(\023)1229 1081 y(AN)28 b(MIN)1559 1060 y(\023)1549 1081 y(A)1620 1060 y(\024)1611 1081 y(C)1671 1051 y FD(\003y)1768 1081 y FE(AND)h(JOHN)e(SW)-9 b(ALLO)n(W)2679 1051 y FD(z)754 1405 y FC(Abstra)n(ct.)42 b FE(In)27= b(our)g(previous)g(pap)r(er)g(w)n(e)h(describ)r(e)f(the)h (Galois)f(mo)r(dule)754 1505 y(structures)37 b(of)h= FB(p)p FE(th-p)r(o)n(w)n(er)e(class)h(groups)f FB(K)2218 1475 y FD(\002)2273 1505 y FB(=3DK)2392 1475 y FD(\002)p FA(p)2481 1505 y FE(,)41 b(where)c FB(K)q(=3DF)49 b FE(is)37 b(a)754 1604 y(cyclic)c(extension)f(of)g(degree)f FB(p)i= FE(o)n(v)n(er)d(a)i (\014eld)h FB(F)44 b FE(con)n(taining)32 b(a)g(primitiv)n(e)754 1704 y FB(p)p FE(th)23 b(ro)r(ot)e(of)h(unit)n(y)-7 b(.)36 b(Our)21= b(description)h(relies)f(up)r(on)i(arithmetic)f(in)n(v)-5 b(arian)n(ts)754 1804 y(asso)r(ciated)19 b(with)i FB(K)q(=3DF)12 b FE(.)34 b(Here)19= b(w)n(e)h(construct)g(\014eld)g(extensions)g FB(K)q(=3DF)31 b FE(with)754 1903 y(prescrib)r(ed)23= b(arithmetic)g(in)n (v)-5 b(arian)n(ts,)22 b(th)n(us)h(completing)g(our)f(classi\014cation) 754 2003 y(of)28 b(Galois)f(mo)r(dules)g FB(K)1506 1973 y FD(\002)1562 2003 y FB(=3DK)1681 1973 y FD(\002)p FA(p)1770 2003 y FE(.)555 2417 y Fz(Let)41 b Fy(F)55 b= Fz(b)s(e)41 b(a)g(\014eld)h(of)e(c)m(haracteristic)j(not)e= Fy(p)f Fz(con)m(taining)i(a)f(primitiv)m(e)h Fy(p)p Fz(th)456 2533 y(ro)s(ot)37 b(of)i(unit)m(y)h Fy(\030)1092 2548 y Fx(p)1131 2533 y Fz(.)62 b(F)-8 b(or)38= b(a)g(cyclic)j(\014eld)e (extension)i Fy(K)r(=3DF)52 b Fz(with)39 b(Galois)g(group)456 2649 y(Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\))39 b(of)g(order)h Fy(p)p Fz(,)i(let)f Fy(J)49 b Fz(=3D)40 b Fy(K)1890 2613 y Fw(\002)1949 2649 y Fy(=3DK)2088 2613 y Fw(\002)p Fx(p)2183 2649 y Fz(,)h(and)f(let)h Fy(N)50 b Fz(denote)41 b(the)f(norm)456 2765 y(map)32 b(from)h Fy(K)39 b Fz(to)33 b Fy(F)14 b Fz(.)555 2975 y(In)37= b([MS)q(],)g(w)m(e)h(pro)m(v)m(ed)g(that)e(the)h (structure)g(of)f(the)h Fv(F)2568 2990 y Fx(p)2608 2975 y Fz([Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\)]-mo)s(dule)456 3091 y Fy(J)41 b Fz(is)33= b(determined)i(b)m(y)e(the)g(follo)m(wing)g (three)g(arithmetic)h(in)m(v)-5 b(arian)m(ts:)679 3334 y Fu(\017)41 b Fy(d)28 b Fz(=3D)f Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(:=3D)g(dim)1610 3349 y Ft(F)1651 3357 y Fs(p)1707 3334 y Fy(F)1784 3298 y Fw(\002)1843 3334 y Fy(=3DN)10 b Fz(\()p Fy(K)2108 3298 y Fw(\002)2167 3334 y Fz(\),)679 3451 y Fu(\017)41 b Fy(e)28 b Fz(=3D)g Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(:=3D)g(dim)1599 3466 y Ft(F)1640 3474 y Fs(p)1696 3451 y Fy(N)10 b Fz(\()p Fy(K)1912 3415 y Fw(\002)1972 3451 y Fz(\))p Fy(=3DF)2136 3415 y Fw(\002)p Fx(p)2230 3451 y Fz(,)33 b(and)679 3569 y Fu(\017)41 b Fz(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(:=3D)32= b(1)h(or)f(0)g(according)h(to)f (whether)i Fy(\030)2552 3584 y Fx(p)2619 3569 y Fu(2)28 b Fy(N)10 b Fz(\()p Fy(K)2929 3533 y Fw(\002)2989 3569 y Fz(\))32 b(or)h(not.)555 3811 y(No)m(w)38 b(if)g Fy(G)e Fz(=3D)g Fv(Z)p Fy(=3Dp)p Fv(Z)p Fz(,)k(then)e Fy(J)46 b Fz(ma)m(y)39 b(b)s(e)f(considered)h(an)f Fv(F)2765 3826 y Fx(p)2804 3811 y Fz([)p Fy(G)p Fz(]-mo)s(dule)g(via)456 3927 y(an)m(y)43 b(isomorphism)i Fy(G)1354 3900 y Fu(\030)1355 3932 y Fz(=3D)1477 3927 y(Gal\()p Fy(K)r(=3DF)14 b= Fz(\),)45 b(and)e(the)h(mo)s(dule)f(structure)i(of)e Fy(J)52 b Fz(is)456 4044 y(indep)s(enden)m(t)29= b(of)e(the)h(c)m(hoice)h(of)e (isomorphism.)43 b(It)28 b(is)g(a)f(fundamen)m(tal)i(problem)456 4160 y(to)41 b(classify)j(the)e(isomorphism)i(classes)f(of)f(mo)s (dules)h Fy(J)51 b Fz(for)41 b(all)h Fy(K)r(=3DF)55 b Fz(in)42 b(our)456 4276 y(con)m(text.)h(This)30= b(problem)f(is)h(solv)m(ed)g(in) f(Theorem)h(1)e(b)s(elo)m(w.)43 b(Corollaries)30 b(1)e(and)456 4392 y(2)k(of)g(Theorem)i(1)e(describ)s(e)i(all)f(mo)s(dules)g Fy(J)42 b Fz(in)33 b(an)f(explicit)j(w)m(a)m(y)-8 b(.)p 456 4526 499 4 v 555 4620 a Fr(Date)6 b FE(:)28 b(April)g(4,)f(2003.) 555 4687 y FD(\003)593 4717 y= FE(Researc)n(h)g(supp)r(orted)h(in)g (part)f(b)n(y)h(the)g(Natural)f(Sciences)h(and)g(Engineering)e(Researc) n(h)456 4817= y(Council)k(of)h(Canada,)g(and)g(b)n(y)f(the)i(sp)r(ecial) e(Dean)h(of)g(Science)g(F)-7 b(und)31= b(at)g(the)h(Univ)n(ersit)n(y)e (of)456 4917 y(W)-7 b(estern)27 b(On)n(tario.)555 4986 y FD(y)590 5016 y= FE(Supp)r(orted)h(b)n(y)f(the)h(Mathematical)f (Sciences)g(Researc)n(h)f(Institute,)j(Berk)n(eley)-7 b(.)555 5086 y FD(z)590 5116 y FE(Researc)n(h)27= b(supp)r(orted)i(in)h (part)e(b)n(y)h(National)g(Securit)n(y)f(Agency)h(gran)n(t)f(MD)n (A904-02-1-)456 5216 y(0061.)1931 5315 y Fq(1)p eop %%Page: 2 2 2 1 bop 456 255 a Fq(2)797 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y= Fz(In)29 b(the)f(follo)m(wing)h(theorem)g(w)m(e)g(determine)h(the)f(sets)g(of)f (in)m(v)-5 b(arian)m(ts)29 b(\()p Fy(d;)17 b(e;)g Fz(\007\))456 566 y(whic)m(h)39 b(ma)m(y)h(b)s(e)e(realized)i(b)m(y)f(an)f(extension) i Fy(K)r(=3DF)52 b Fz(and)38 b(in)h(so)f(doing)h(classify)456 683 y(all)32 b Fv(F)652 698 y Fx(p)692 683 y Fz([)p Fy(G)p Fz(]-mo)s(dules)h Fy(J)k Fz(=3D)27 b Fy(J)9 b Fz(\()p Fy(K)r(=3DF)14 b Fz(\))32 b(up)h(to)f(isomorphism.)456 859 y FF(Theorem)g(1.)37 b Fp(L)-5 b(et)31 b Fy(p)f Fp(b)-5 b(e)29 b(a)h(prime)g(numb)-5 b(er.)42 b(F)-7 b(or)30 b(arbitr)-5 b(ary)30 b(c)-5= b(ar)g(dinal)29 b(numb)-5 b(ers)456 975 y Fy(d)p Fp(,)40 b Fy(e)p= Fp(,)i(and)d(for)h Fz(\007)d Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)p Fp(,)40 b(ther)-5 b(e)40 b(exists)g(a)f(cyclic)h(\014eld)f(extension)g Fy(K)r(=3DF)54 b Fp(of)456 1092 y(de)-5 b(gr)g(e)g(e)25 b Fy(p)g Fp(c)-5 b(ontaining)25 b(a)g(primitive)g Fy(p)p Fp(th)h(r)-5 b(o)g(ot)25 b(of)h(unity)g(with)g(invariants)e Fz(\()p Fy(d;)17 b(e;)g Fz(\007\))456 1208 y Fp(if)34 b(and)h(only)f(if)679 1431 y Fu(\017)41 b Fp(if)35 b Fz(\007)27 b(=3D)h(0)p Fp(,)34 b(then)h Fz(1)28 b Fu(\024)g Fy(d)p Fp(,)679 1547 y Fu(\017)41 b Fp(if)35 b Fy(p)27 b(>)h Fz(2)35 b Fp(then)f Fz(1)28 b Fu(\024)g Fy(e)p Fp(,)35 b(and)679 1663 y Fu(\017)41 b Fp(if)35 b Fy(p)27 b Fz(=3D)h(2)35 b Fp(and)f Fz(\007)27 b(=3D)h(1)35 b Fp(then)f Fz(1)28 b Fu(\024)g Fy(e)p Fp(.)555 1925 y Fz(F)-8= b(rom)29 b(the)i(theorem)f(ab)s(o)m(v)m(e)h(and)f(from)f([MS)q(,)h(Theorem)h(3)f (and)g(Corollary)g(2])456 2041= y(w)m(e)e(immediately)g(obtain)f(the)g (follo)m(wing)h(corollaries.)42 b(W)-8 b(e)27 b(denote)h(b)m(y)g Fy(M)3202 2056 y Fx(i;j)3309 2041 y Fz(the)456 2157 y Fy(j)6 b Fz(th)30 b(cyclic)j(mo)s(dule)e Fv(F)1294 2172 y Fx(p)1334 2157 y Fz([)p Fy(G)p Fz(])g(suc)m(h)h(that)e(dim)2086 2172 y Ft(F)2127 2180 y Fs(p)2184 2157 y Fy(M)2278 2172 y Fx(i;j)2386 2157 y Fz(=3D)d Fy(i)p Fz(,)32 b(where)g Fy(j)k Fz(is)c(a)e(suitable)456 2273 y(index.)456 2450 y FF(Corollary)i(1.)37 b Fp(L)-5 b(et)29 b Fy(p)f(>)f Fz(2)i Fp(b)-5 b(e)29 b(a)g(prime)f(numb)-5 b(er,)30 b(and)f(let)g Fy(G)g Fp(b)-5 b(e)29 b(a)g(cyclic)g(gr)-5 b(oup)456 2566 y(of)33 b(or)-5 b(der)33 b Fy(p)p Fp(.)45 b(Then)33 b(an)g Fv(F)1400 2581 y Fx(p)1440 2566 y Fz([)p Fy(G)p Fz(])p Fp(-mo)-5 b(dule)33 b Fy(J)43 b Fp(is)33 b(r)-5 b(e)g(alizable)33 b(as)g(an)g Fv(F)2877 2581 y Fx(p)2917 2566 y Fz([Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\)])p Fp(-)456 2682 y(mo)-5 b(dule)40 b Fy(K)883 2646 y Fw(\002)942 2682 y Fy(=3DK)1081 2646 y Fw(\002)p Fx(p)1217 2682 y= Fp(for)h(some)g (cyclic)g Fy(G)p Fp(-extension)f Fy(K)r(=3DF)54 b Fp(such)41 b(that)h Fy(F)55 b Fp(c)-5 b(on-)456 2799 y(tains)43 b(a)h(primitive)g Fy(p)p Fp(th)g(r)-5 b(o)g(ot)44= b(of)g(unity)h(if)f (and)g(only)f(if)h(ther)-5 b(e)44 b(exist)g(c)-5 b(ar)g(dinal)456 2915 y(numb)g(ers)34 b Fy(d)p Fp(,)g Fy(e)p Fp(,)h(and)g Fz(\007)27 b Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)34 b Fp(such)h(that)620 3138 y Fz(\()p Fy(i)p Fz(\))41 b Fp(If)34 b Fz(\007)28 b(=3D)g(0)p Fp(,)34 b(then)h Fz(1)27 b Fu(\024)h Fy(d)p Fp(;)586 3254 y Fz(\()p Fy(ii)p Fz(\))42 b(1)28 b Fu(\024)g Fy(e)p Fp(;)35 b(and)553 3370 y Fz(\()p Fy(iii)p Fz(\))858 3652 y Fy(J)h Fz(=3D)1052 3482 y Fo( )1136 3558 y(M)1131 3770 y Fx(j)t Fw(2)p Fn(K)1257 3779 y Fm(1)1308 3652 y Fy(M)1402 3667 y Fl(1)p Fx(;j)1493 3482 y Fo(!)1589 3558 y(M)1756 3482 y( )1840 3558 y(M)1835 3770 y Fx(j)t Fw(2)p Fn(K)1961 3779 y Fm(2)2012 3652 y Fy(M)2106 3667 y Fl(2)p Fx(;j)2197 3482 y Fo(!)2293 3558 y(M)2460 3452 y(0)2460 3632 y(@)2553 3558 y(M)2547 3770 y Fx(j)t Fw(2)p Fn(K)2673 3778 y Fs(p)2725 3652 y Fy(M)2819 3667 y Fx(p;j)2911 3452 y Fo(1)2911 3632 y(A)3015 3652 y Fy(;)770 3929 y Fp(wher)-5 b(e)803 4090 y Fz(\(1\))41 b Fu(j)p Fk(K)1064 4105 y Fl(1)1103 4090 y Fu(j)21 b Fz(+)h(1)28 b(=3D)f(2\007)22 b(+)g Fy(d)p Fp(,)803 4206 y Fz(\(2\))41 b Fu(j)p Fk(K)1064 4221 y Fl(2)1103 4206 y Fu(j)27 b Fz(=3D)g(1)22 b Fu(\000)h Fz(\007)p Fp(,)35 b(and)802 4322 y Fz(\()p Fy(p)p Fz(\))42 b Fu(j)p Fk(K)1064 4337 y Fx(p)1103 4322 y Fu(j)22 b Fz(+)g(1)27 b(=3D)h Fy(e)p Fp(.)456 4545 y(The)34 b(invariants)g Fy(d)p Fp(,)g Fy(e)p Fp(,)h(and)g Fz(\007)f= Fp(determine)g(the)h(mo)-5 b(dule)35 b Fy(J)44 b Fp(uniquely.)555 4807 y Fz(F)-8 b(or)38 b Fy(p)h= Fz(=3D)f(2)h(using)h([MS,)h(Theorem)f(3)f(and)g (Corollary)g(3],)h(along)f(with)g(our)456 4923= y(theorem)33 b(ab)s(o)m(v)m(e,)g(w)m(e)h(obtain)f(the)g(next)g(corollary)-8 b(.)456 5099 y FF(Corollary)38 b(2.)j Fp(Now)35 b(let)f Fy(G)g Fp(b)-5 b(e)34 b(a)g(cyclic)g(gr)-5 b(oup)34 b(of)g(or)-5 b(der)34 b Fz(2)p Fp(.)45 b(Then)33 b(an)h Fv(F)3239 5114 y Fl(2)3278 5099 y Fz([)p Fy(G)p Fz(])p Fp(-)456 5216 y(mo)-5 b(dule)37 b Fy(J)46 b Fp(is)38 b(r)-5= b(e)g(alizable)36 b(as)i(an)f Fv(F)1751 5231 y Fl(2)1790 5216 y Fz([Gal\()p Fy(K)r(=3DF)14 b Fz(\)])p Fp(-mo)-5 b(dule)36 b Fy(K)2742 5179 y Fw(\002)2801 5216 y Fy(=3DK)2940 5179 y Fw(\002)p Fl(2)3072 5216 y Fp(for)h(some)p eop %%Page: 3 3 3 2 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)330 b(3)456 450 y Fp(quadr)-5 b(atic)24 b(extension)g Fy(K)r(=3DF)38 b Fp(with)25 b(its)g(arithmetic)g (invariants)f Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))p Fp(,)25 b Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))p Fp(,)456 566 y(and)35 b Fz(\007\()p Fy(K)r(=3DF)14 b Fz(\))36 b Fp(c)-5 b(oinciding)34 b(with)j Fy(d)p Fp(,)f Fy(e)p Fp(,)h(and)e Fz(\007)c Fu(2)f(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)p Fp(,)36 b(r)-5 b(esp)g(e)g(ctively,)36 b(if)g(and)456 683 y(only)e(if)h Fy(d)p Fp(,)f Fy(e)p Fp(,)h(and)g Fz(\007)f= Fp(satisfy)h(the)g(c)-5 b(onditions)34 b(b)-5 b(elow.)679 898 y Fu(\017)41 b Fp(if)35 b Fz(\007)27 b(=3D)h(0)p Fp(,)34 b(then)h Fz(1)28 b Fu(\024)g Fy(d)p Fp(,)679 1015 y Fu(\017)41 b Fp(if)35 b Fz(\007)27 b(=3D)h(1)p Fp(,)34 b(then)h Fz(1)28 b Fu(\024)g Fy(e)p Fp(.)555 1231 y(In)34 b(this)h(c)-5 b(ase)1241 1454 y Fy(J)37 b Fz(=3D)1435 1284 y Fo( )1519 1359 y(M)1514 1571 y Fx(j)t Fw(2)p Fn(K)1640 1580 y Fm(1)1691 1454 y Fy(M)1785 1469 y Fl(1)p Fx(;j)1876 1284 y Fo(!)1972 1359 y(M)2139 1284 y( )2223 1359 y(M)2218 1571 y Fx(j)t Fw(2)p Fn(K)2344 1580 y Fm(2)2395 1454 y Fy(M)2489 1469 y Fl(2)p Fx(;j)2580 1284 y Fo(!)456 1712 y Fp(wher)-5 b(e)604 1928 y Fz(\(1\))41 b Fu(j)p Fk(K)865 1943 y Fl(1)904 1928 y Fu(j)22 b Fz(+)g(1)27 b(=3D)h(2\007)21 b(+)h Fy(d)35 b Fp(and)604 2045 y Fz(\(2\))41 b Fu(j)p Fk(K)865 2060 y Fl(2)904 2045 y Fu(j)22 b Fz(+)g(\007)27 b(=3D)h Fy(e)p Fp(.)555 2261 y(Mor)-5 b(e)g(over,)34 b(such)f(a)g(mo)-5 b(dule)33 b Fy(J)43 b Fp(is)33 b(determine)-5= b(d)32 b(uniquely)i(by)g(the)f(invariants)456 2377 y Fz(2\007)21 b(+)h Fy(d)35 b Fp(and)f(by)h Fy(e)23 b Fu(\000)f Fz(\007)35 b Fp(if)g Fy(e)g Fp(is)f(\014nite)h(and)f(by)h Fy(e)g Fp(alone)f(if)h Fy(e)g Fp(is)g(in\014nite.)555 2625 y Fz(If)g Fy(p)d(>)g Fz(2)j(then)h(t)m(w)m(o)g Fv(F)1401 2640 y Fx(p)1441 2625 y Fz([)p Fy(G)p= Fz(]-mo)s(dules)f(are)h (isomorphic)g(if)f(and)g(only)h(if)f(their)456 2742= y(in)m(v)-5 b(arian)m(ts)31 b Fy(d)p Fz(,)f Fy(e)p= Fz(,)h(and)f(\007)f(are)h(the)h (same,)g(b)m(y)g([MS)q(,)f(Corollary)g(2].)43 b(Th)m(us)31 b(w)m(e)g(see)456 2858 y(in)i(particular)h(that)f(if)h Fy(p)29 b(>)g Fz(2,)34 b(then)g(the)g(arithmetic)g(in)m(v)-5 b(arian)m(ts)35 b(of)e Fy(J)42 b Fz(dep)s(end)456 2974 y(only)33 b(up)s(on)f(the)h(isomorphism)h(t)m(yp)s(e)g(of)e(the)h Fv(F)2214 2989 y Fx(p)2254 2974 y Fz([)p Fy(G)p Fz(]-mo)s(dule)f Fy(J)9 b Fz(.)555 3170 y(In)45 b(the)h(case)g= Fy(p)i Fz(=3D)h(2,)f(w)m(e)d(see)i(from)d([MS)q(,)k(Corollary)d(2])g(again)f (that)h(the)456 3286 y(arithmetic)29 b(in)m(v)-5 b(arian)m(ts)30 b Fy(d)p Fz(,)g Fy(e)p Fz(,)g(and)f(\007)f(determine)j(our)e(mo)s(dule) g Fv(F)2890 3301 y Fl(2)2930 3286 y Fz([)p Fy(G)p= Fz(],)g(but)h(t)m(w)m (o)456 3402 y(isomorphic)d Fv(F)1004 3417 y Fl(2)1043 3402 y Fz([)p Fy(G)p= Fz(]-mo)s(dules)g(ma)m(y)f(ha)m(v)m(e)i (di\013eren)m(t)f(arithmetic)g(in)m(v)-5 b(arian)m(ts;)29 b(see)456 3518 y([MS,)j(Corollary)g(3].)43= b(Here)33 b(is)f(a)g(v)m(ery)h(simple,)h(concrete)e(example)i(illustrating)456 3635 y(this)46 b(p)s(ossibilit)m(y)-8 b(.)87 b(Let)46 b Fy(K)1466 3650 y Fl(1)1506 3635 y Fy(=3DF)1618 3650 y Fl(1)1703 3635 y= Fz(b)s(e)g(a)g(quadratic)h(extension)h(of)e(\014nite)h (\014elds)456 3751 y(of)e(c)m(haracteristic)j(not)e(2,)k Fy(F)1569 3766 y Fl(2)1659 3751 y Fz(=3D)h Fv(R)p Fz(\(\()p Fy(t)p Fz(\)\))46= b(b)s(e)g(a)g(\014eld)h(of)f(p)s(o)m(w)m(er)h(series) h(with)456 3870 y(co)s(e\016cien)m(ts)26 b(in)e(real)g(n)m(um)m(b)s (ers)i Fv(R)p Fz(,)f(and)g Fy(K)2001 3885 y Fl(2)2068 3870 y Fz(=3D)i Fy(F)2234 3885 y Fl(2)2274 3870 y Fz(\()2312 3792 y Fu(p)p 2395 3792 127 4 v 78 x(\000)p Fz(1\).)40 b(Then)26 b(b)s(oth)d(mo)s(dules)456 3988 y Fy(K)546 3946 y Fw(\002)539 4012 y Fl(1)605 3988 y Fy(=3DK)744 3946 y Fw(\002)p Fl(2)737 4012 y(1)861 3988 y Fz(and)g Fy(K)1131 3946 y Fw(\002)1124 4012 y Fl(2)1191 3988 y Fy(=3DK)1330 3946 y Fw(\002)p Fl(2)1323 4012 y(2)1447 3988 y= Fz(are)g(isomorphic)i (to)d(a)h(trivial)h Fv(F)2608 4003 y Fl(2)2647 3988 y Fz([)p Fy(G)p Fz(]-mo)s(dule)g Fv(F)3207 4003 y Fl(2)3246 3988 y Fz(,)h(but)456 4104 y(their)31= b(arithmetic)h(in)m(v)-5 b(arian)m(ts)32 b(\()p Fy(d)1692 4119 y Fx(i)1720 4104 y Fy(;)17 b(e)1809 4119 y Fx(i)1837 4104 y Fy(;)g Fz(\007)1957 4119 y Fx(i)1985 4104 y Fz(\))31 b(are)g(\(0)p Fy(;)17 b Fz(1)p Fy(;)g Fz(1\))29 b(for)i Fy(i)c Fz(=3D)h(1)j(and)g(\(2)p Fy(;)17 b Fz(0)p Fy(;)g Fz(0\))456 4220 y(for)32 b Fy(i)c Fz(=3D)f(2.)1289 4497 y(1.)48 b Fj(Not)-7 b(a)g(tion)40 b(and)e(Stra)-7 b(tegy)555 4751 y Fz(In)29= b(all)g(that)g(follo)m(ws)h Fy(F)42 b Fz(denotes)30 b(a)f(\014eld,)h Fy(F)2174 4714 y Fw(\002)2261 4751 y Fz(=3D)d Fy(F)i Fu(n)15 b(f)p Fz(0)p Fu(g)27 b Fz(the)j(m)m(ultiplicativ)m(e)456 4867 y(group)39 b(of)g Fy(F)14 b Fz(,)40 b Fy(p)g Fz(a)f(prime)h(n)m(um)m(b)s(er,)i (and)e Fy(F)2125 4831 y Fw(\002)2184 4867 y Fy(=3DF)2310 4831 y Fw(\002)p Fx(p)2443 4867 y Fz(the)g(group)f(of)g Fy(p)p Fz(th-p)s(o)m(w)m(er)456 4983 y(classes)47 b(of)f Fy(F)14 b Fz(.)83 b(F)-8 b(or)45 b(eac)m(h)i Fy(f)61 b Fu(2)50 b Fy(F)1814 4947 y Fw(\002)1919 4983 y= Fz(w)m(e)d(denote)f(b) m(y)h([)p Fy(f)11 b Fz(])46 b(the)g(class)h(of)f Fy(f)56 b Fz(in)456 5099 y Fy(F)533 5063 y Fw(\002)591 5099 y Fy(=3DF)717 5063 y Fw(\002)p Fx(p)811 5099 y Fz(.)48 b(F)-8 b(or)33 b(eac)m(h)i(subset)g Fy(A)f Fz(of)g Fy(F)1880 5063 y Fw(\002)1972 5099 y Fz(w)m(e)h(denote)g(b)m(y)g([)p Fy(A)p Fz(])f(the)g(set)h(of)e(classes)456 5216 y Fu(f)p Fz([)p Fy(a)p Fz(])f Fu(j)h Fy(a)27 b Fu(2)h Fy(A)p Fu(g)33 b Fz(and)g(b)m(y)g Fu(h)p Fz([)p Fy(A)p Fz(])p= Fu(i)f Fz(the)h(subgroup)h(of)e Fy(F)2374 5179 y Fw(\002)2433 5216 y Fy(=3DF)2559 5179 y Fw(\002)p Fx(p)2685 5216 y= Fz(generated)h(b)m (y)h([)p Fy(A)p Fz(].)p eop %%Page: 4 4 4 3 bop 456 255 a Fq(4)797 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y= Fz(W)h(e)29 b(denote)h(b)m(y)g Fy(\030)1205 465 y Fx(p)1273 450 y Fz(a)e(primitiv)m(e)j Fy(p)p Fz(th)d(ro)s(ot)g(of)h(unit)m(y)g(in)g Fy(F)14 b Fz(.)42 b(\(Some)30 b(\014elds)g(will)456= 566 y(b)s(e)g(assumed)i(to)e(con)m(tain)h(suc)m(h)g(a)f(primitiv)m(e)i Fy(p)p Fz(th)f(ro)s(ot;)f(for)g(the)g(other)h(\014elds)g(in)456 683= y(this)k(pap)s(er,)g(w)m(e)h(will)f(pro)m(v)m(e)h(that)e(a)g (primitiv)m(e)i Fy(p)p Fz(th)f(ro)s(ot)e(is)j(con)m(tained)f(in)g(the) 456 799 y(\014eld.\))58 b(Observ)m(e)39= b(that)e(our)g(assumption)i (that)e(there)h(exists)h(a)e(primitiv)m(e)i Fy(p)p Fz(th)456 915 y(ro)s(ot)31 b(of)h(unit)m(y)i(implies)g(that)f(c)m(har\()p Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h Fy(p)p Fz(.)555 1118 y(F)-8 b(or)43 b(a)f(Galois)h(extension)i Fy(K)r(=3DF)14 b Fz(,)45 b(Gal\()p Fy(K)r(=3DF)14 b Fz(\))42= b(denotes)i(the)g(Galois)f (group)456 1234 y(and)34 b Fy(N)725 1249 y Fx(K)q(=3DF)913 1234 y Fz(denotes)h(the)g(norm)f(map)g(from)g Fy(K)41 b Fz(to)34 b Fy(F)14 b Fz(.)47 b(W)-8 b(e)35 b(denote)g(b)m(y)f Fy(F)3238 1198 y Fx(s)3309 1234 y Fz(the)456 1350= y(separable)j (closure)g(of)e Fy(F)50 b Fz(and)35 b Fy(G)1713 1365 y Fx(F)1808 1350 y Fz(the)h(absolute)h(Galois)e(group)h(Gal)o(\()p Fy(F)3217 1314 y Fx(s)3254 1350 y Fy(=3DF)14 b Fz(\).)456 1466 y(As)43 b(usual,)j Fy(H)994 1430 y Fx(i)1022 1466 y Fz(\()p Fy(G)1137 1481 y Fx(F)1195 1466 y Fy(;)17 b Fv(F)1300 1481 y Fx(p)1339 1466 y Fz(\))43= b(are)g(Galois)f(cohomology) h(groups)g(of)f Fy(F)57 b Fz(with)43 b(co)s(ef-)456= 1582 y(\014cien)m(ts)g(in)e Fv(F)976 1597 y Fx(p)1016 1582 y Fz(.)69 b(Since)43= b(all)e(absolute)h(Galois)f(groups)h(will)g(b)s(e) f(pro-)p Fy(p)p Fz(-groups,)456 1699 y(all)c(considered)j Fv(F)1140 1714 y Fx(p)1217 1699 y= Fz(mo)s(dules)f(are)f(trivial.)59 b(Finally)-8 b(,)40 b(let)e Fu(j)p Fy(B)5 b Fu(j)37= b Fz(b)s(e)h(the)g(cardinal)456 1815 y(n)m(um)m(b)s(er)c(of)e(a)g(set)h Fy(B)5 b Fz(.)555 2017 y(First)26= b(observ)m(e)i(that)d(the)i (conditions)f(on)g Fy(d)p Fz(,)h Fy(e)p= Fz(,)g(and)f(\007)g(listed)h (in)f(our)g(theorem)456 2134 y(ab)s(o)m(v)m(e)33 b(are)g(necessary:)604 2360 y(\(1\))41 b(If)33 b(\007)27 b(=3D)h(0,)k(then)h Fy(\030)1448 2375 y Fx(p)1527 2360 y Fy(=3D)-61 b Fu(2)28 b Fy(N)1687 2376 y Fx(K)q(=3DF)1842 2360 y Fz(\()p Fy(K)1970 2324 y Fw(\002)2029 2360 y Fz(\))k(and)h(hence)1306 2531 y Fy(d)28 b Fz(=3D)f(dim)1651 2546 y Ft(F)1692 2554 y Fs(p)1748 2531 y Fy(F)1825 2489 y Fw(\002)1884 2531 y Fy(=3D)-5 b(N)2006 2546 y Fx(K)q(=3DF)2160 2531 y Fz(\()p Fy(K)2288 2489 y Fw(\002)2347 2531 y Fz(\))28 b Fu(\025)g Fz(1;)604 2700 y(\(2\))41 b(If)g Fy(p)h(>)f Fz(2)g(and)g Fy(K)48 b Fz(=3D)42 b Fy(F)14 b Fz(\()1763 2667 y Fs(p)1740 2628 y Fu(p)p 1823 2628 52 4 v 72 x Fy(a)p Fz(\))41 b(for)f(a)h(suitable)h Fy(a)f Fu(2)i Fy(F)2850 2664 y Fw(\002)2936 2700 y Fu(n)28 b Fy(F)3091 2664 y Fw(\002)p Fx(p)3185 2700 y Fz(,)43 b(then)770 2816 y Fy(a)28 b Fz(=3D)g Fy(N)1031 2832 y Fx(K)q(=3DF)1185 2816 y Fz(\()1250 2783 y Fs(p)1227 2745 y Fu(p)p 1310 2745 V 71 x Fy(a)p Fz(\))33 b(and)g(hence)1400 2987 y Fy(e)27 b Fz(=3D)h(dim)1739 3002 y Ft(F)1780 3010 y Fs(p)1836 2987 y Fy(N)1914 3002 y Fx(K)q(=3DF)2068 2987 y Fz(\()p Fy(K)2196 2945 y Fw(\002)2255 2987 y Fz(\))g Fy(>)g Fz(0)p Fy(:)604 3156 y Fz(\(3\))41 b(If)j Fy(p)k Fz(=3D)g(2,)f(\007)g(=3D)h(1,)f (and)d Fy(K)55 b Fz(=3D)48 b Fy(F)14 b Fz(\()2171 3084 y Fu(p)p 2253 3084 V 2253 3156 a Fy(a)p Fz(\))45= b(for)e(a)h(suitable)i Fy(a)i Fu(2)g Fy(F)3306 3120 y Fw(\002)3395 3156 y Fu(n)770 3272 y Fy(F)847 3236 y Fw(\002)p Fl(2)941 3272 y Fz(,)d(then)f Fu(\000)p Fz(1)h Fu(2)g Fy(N)1606 3288 y Fx(K)q(=3DF)1760 3272 y Fz(\()p Fy(K)1888 3236 y Fw(\002)1947 3272 y= Fz(\))e(since)h (\007)g(=3D)h(1.)73 b(Consequen)m(tly)46 b Fy(a)f Fu(2)770 3389 y Fy(N)848 3405 y Fx(K)q(=3DF)1002 3389 y Fz(\()p Fy(K)1130 3353 y Fw(\002)1190 3389 y Fz(\),)32 b(and)h(th)m(us)g(1)28 b Fu(\024)g Fy(e)p Fz(.)456 3616 y(Therefore)38= b(in)f(order)g(to)f (pro)m(v)m(e)j(Theorem)f(1)e(when)j Fy(p)c(>)f= Fz(2,)k(it)f(is)h (su\016cien)m(t)h(to)456 3732 y(sho)m(w,)30= b(for)d(eac)m(h)i(cardinal) g(n)m(um)m(b)s(ers)g Fy(d)p Fz(,)g Fy(e)f= Fz(as)g(ab)s(o)m(v)m(e,)i (for)e(eac)m(h)h(\007)e Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)p Fz(,)28 b(and)456 3848 y(for)k(eac)m(h)h(prime)h(n)m(um)m(b)s (er)f Fy(p)28 b(>)g Fz(2,)k(the)h(existence)i(of)d(a)h(\014eld)g Fy(F)46 b Fz(suc)m(h)34 b(that:)679 4075 y Fu(\017)41 b Fy(F)46 b Fz(con)m(tains)34 b(a)e(primitiv)m(e)j Fy(p)p Fz(th)d(ro)s(ot)g Fy(\030)2186 4090 y Fx(p)2225 4075 y Fz(;)679 4191 y Fu(\017)41 b Fy(F)847 4155 y Fw(\002)906 4191 y Fy(=3DF)1032 4155 y Fw(\002)p Fx(p)1158 4191 y= Fz(decomp)s(oses)35 b(in)m(tro)e(a)f(direct)h(sum)h(of)e(subgroups)1399 4356 y Fy(F)1476 4315 y Fw(\002)1535 4356 y Fy(=3DF)1661 4315 y Fw(\002)p Fx(p)1782 4356 y Fz(=3D)c Fy(D)d Fu(\010)d(h)p Fz([)p Fy(a)p Fz(])p Fu(i)g(\010)h Fy(E)6 b(;)770 4521 y Fz(where)34 b(dim)1215 4536 y Ft(F)1256 4544 y Fs(p)1295 4521 y Fz(\()p Fu(h)p Fz([)p Fy(a)p Fz(])p Fu(i)22 b(\010)h Fy(E)6 b Fz(\))28 b(=3D)f Fy(e)33 b Fz(and,)g(setting)g Fy(K)i Fz(=3D)27 b Fy(F)14 b Fz(\()2864 4487 y Fs(p)2840 4449 y Fu(p)p 2923 4449 V 72 x Fy(a)q Fz(\),)803 4638 y(\(1\))41 b([)p Fy(N)1074 4654 y Fx(K)q(=3DF)1228 4638 y Fz(\()p Fy(K)1356 4602 y Fw(\002)1415 4638 y Fz(\)])28 b(=3D)g Fu(h)p Fz([)p Fy(a)p Fz(])p Fu(i)22 b(\010)g Fy(E)6 b Fz(;)803 4755 y(\(2\))41 b(dim)1131 4770 y Ft(F)1172 4778 y Fs(p)1212 4755 y Fz(\()p Fy(F)1327 4719 y Fw(\002)1386 4755 y Fy(=3D)-5 b(N)1508 4771 y Fx(K)q(=3DF)1662 4755 y Fz(\()p Fy(K)1790 4719 y Fw(\002)1849 4755 y Fz(\)\))27 b(=3D)h(dim)2218 4770 y Ft(F)2259 4778 y Fs(p)2316 4755 y Fy(D)i Fz(=3D)e Fy(d)p Fz(;)k(and)803 4873 y(\(3\))41 b(\007)27 b(=3D)h(0)k(if)h(and)f(only)h(if)g Fy(\030)1883 4888 y Fx(p)1961 4873 y Fy(=3D)-60 b Fu(2)28 b Fy(N)2122 4888 y Fx(K)q(=3DF)2276 4873 y Fz(\()p Fy(K)2404 4836 y Fw(\002)2463 4873 y Fz(\).)555 5099 y(In)37 b(the)g(case)g Fy(p)d= Fz(=3D)g(2)i(and)h(\007)d(=3D)g(1)i(w)m(e)h(use)g(the)g(same)g (conditions)h(as)e(ab)s(o)m(v)m(e,)456 5216 y(and)45 b(if)f Fy(p)49 b Fz(=3D)f(2)d(and)g(\007)j(=3D)h(0)44= b(w)m(e)i(require)g (instead)g(that)f Fy(e)j Fz(=3D)h(dim)3068 5231 y Ft(F)3109 5240 y Fm(2)3164 5216 y Fy(E)i Fz(and)p eop %%Page: 5 5 5 4 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)330 b(5)456 450 y Fy(d)35 b Fz(=3D)h(dim)816 465 y Ft(F)857 474 y Fm(2)896 450 y Fz(\()p Fy(D)28 b Fu(\010)e(h)p Fz([)p Fy(a)p Fz(])p Fu(i)p Fz(\).)57= b(The)39 b(latter)e(condition)h(is)g(imp)s(osed)g(b)s(ecause)h Fu(\000)p Fz(1)48 b Fy(=3D)-61 b Fu(2)456 566 y Fy(N)534 586 y Fx(F)10 b Fl(\()616 538 y Fw(p)p 674 538 38 3 v 674 586 a Fx(a)q Fl(\))p Fx(=3DF)833 566 y Fz(\()p Fy(F)k Fz(\()986 495 y Fu(p)p 1069 495 52 4 v 71 x Fy(a)p Fz(\))1158 530 y Fw(\002)1217 566 y Fz(\))32 b(if)h(and)g(only)g(if)f Fy(a)40 b(=3D)-61 b Fu(2)28 b Fy(N)2121 586 y Fx(F)10 b Fl(\()2203 538 y Fw(p)p 2262 538 38 3 v 48 x Fx(a)p Fl(\))p Fx(=3DF)2421 566 y Fz(\()p Fy(F)k Fz(\()2574 495 y Fu(p)p 2656 495 52 4 v 2656 566 a Fy(a)p Fz(\))2745 530 y Fw(\002)2804 566 y Fz(\).)555 769 y(Our)23= b(strategy)h(is)g(to)f(in)m(terpret)h (the)g(required)g(conditions)h(on)e Fy(F)2899 733 y Fw(\002)2957 769 y Fy(=3DF)3083 733 y Fw(\002)p Fx(p)3201 769 y Fz(ab)s(o)m(v)m(e)456 886 y(in)32 b(terms)h(of)f(Galois)g(cohomology)-8 b(.)43 b(W)-8 b(e)33 b(then)g(observ)m(e)h(that)e(these)h(conditions)456 1002 y(are)39 b(satis\014ed)i(if)e Fy(G)1180 1017 y Fx(F)1278 1002 y Fz(is)h(a)f(free)h(pro)s(duct,)i(in)d(the)h(category)g(of)f (pro-)p Fy(p)p Fz(-groups,)456 1118 y(of)f(suitable)j(pro-)p Fy(p)p Fz(-groups)d Fy(G)1597 1133 y Fl(1)1676 1118 y Fz(and)h Fy(G)1949 1133 y Fl(2)1989 1118 y= Fz(,)i(and)e(\014nally)h(w)m (e)h(use)f(the)g(v)m(ery)g(nice)456 1234= y(theorem)32 b(pro)m(v)m(ed)h(b)m(y)g(Efrat)f(and)g(Haran)f(whic)m(h)i(guaran)m (tees)g(the)f(existence)j(of)456 1351 y(a)28 b(\014eld)i(with)g Fy(G)1037 1366 y Fx(F)1124 1351 y Fz(ab)s(o)m(v)m(e.)43 b(This)30= b(is)g(one)f(of)g(the)g(k)m(ey)i(results)f(used)g(in)f(our)g (pap)s(er.)456 1524 y FF(Theorem)42 b(2.)i Fp(\(Efr)-5 b(at-Har)g(an;)40 b(se)-5 b(e)38 b= Fz([EH)q(,)f(Prop)s(osition)g(1.3])p Fp(\))h(L)-5 b(et)39 b Fy(F)3076 1539 y Fl(1)3116 1524 y Fy(;)17 b(:)g(:)g(:)f(;)h(F)3398 1539 y Fx(n)456 1640 y Fp(b)-5 b(e)26 b(\014elds)g(of)h(e)-5 b(qual)26 b(char)-5 b(acteristic)27 b(such)f(that)i Fy(G)2233 1655 y Fx(F)2278 1664 y Fm(1)2316 1640 y Fy(;)17 b(:)g(:)g(:)f(;)h(G)2612 1655 y Fx(F)2657 1663 y Fs(n)2730 1640 y Fp(ar)-5 b(e)26 b(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)456 1756 y(Then)34 b(ther)-5 b(e)34 b(exists)h(a)g(\014eld)f Fy(F)48 b= Fp(of)35 b(the)g(same)f(char)-5 b(acteristic)34 b(such)h(that)1487 1913 y Fy(G)1564 1928 y Fx(F)1650 1886 y Fu(\030)1651 1918 y Fz(=3D)1755 1913 y Fy(G)1832 1928 y Fx(F)1877 1937 y Fm(1)1938 1913 y Fy(?)22 b Fu(\001)17 b(\001)g(\001)j Fy(?)i(G)2295 1928 y Fx(F)2340 1936 y Fs(n)2386 1913 y Fy(;)456 2070 y Fp(wher)-5 b(e)34 b(the)h(pr)-5 b(o)g(duct)35 b(is)f(fr)-5 b(e)g(e)35 b(in)f(the)h(c)-5 b(ate)g(gory)35 b(of)g(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)456 2326 y Fz(In)35= b(order)f(to)h(apply)g(this)h(theorem,)g(w)m(e)f(sho)m(w)h (the)f(existence)j(of)c(the)h(\014elds)h Fy(F)3405 2341 y Fl(1)456 2442 y Fz(and)27 b Fy(F)703 2457 y Fl(2)769 2442 y Fz(suc)m(h)i(that)e Fy(G)1267 2457 y Fx(F)1312 2466 y Fm(1)1377 2442 y Fz(and)h Fy(G)1639 2457 y Fx(F)1684 2466 y Fm(2)1749 2442 y Fz(are)f(prescrib)s(ed)i(Galois)e(groups)g Fy(G)3050 2457 y Fl(1)3117 2442 y Fz(and)g Fy(G)3378 2457 y Fl(2)3417 2442 y Fz(.)456 2559 y(W)-8 b(e)33= b(use)g(the)g(tec)m (hniques)i(of)d(henselian)j(v)-5 b(aluations)32= b(and)h(formal)f(p)s(o) m(w)m(er)i(series)456 2675 y(to)e(construct)i(\014elds)f Fy(F)1316 2690 y Fl(1)1388 2675 y Fz(and)g Fy(F)1641 2690 y Fl(2)1681 2675 y Fz(.)1698 2955 y(2.)49 b Fj(Lemmas)456 3211 y Fz(2.1.)f FF(V)-9 b(alued)41 b(\014elds)g Fy(F)1387 3226 y Fl(1)1466 3211 y= FF(with)f(prescrib)s(ed)h(residue)f(\014eld)h Fy(F)2970 3226 y Fl(0)3050 3211 y FF(and)f(v)-6 b(al-)456 3328 y(uation)38 b(group)f Fz(\000)p FF(.)555 3526 y Fz(Let)48 b Fy(v)j Fz(b)s(e)d(a)f(v)-5 b(aluation)47 b(on)g(a)h(\014eld)g Fy(F)2063 3541 y Fl(1)2102 3526 y Fz(,)k(written)c(additiv)m(ely)-8 b(.)90 b(Then)48 b(w)m(e)456 3642 y(denote)28 b(b)m(y)g Fy(A)968 3657 y Fx(v)1037 3642 y Fz(the)g(v)-5 b(aluation)27 b(ring)g Fu(f)p Fy(f)39 b Fu(2)28 b Fy(F)2109 3657 y Fl(1)2176 3642 y Fu(j)f Fy(v)t Fz(\()p Fy(f)11 b Fz(\))27 b Fu(\025)h Fz(0)p Fu(g)p Fz(;)h(b)m(y)f Fy(M)2928 3657 y Fx(v)2997 3642 y Fz(the)g(unique)456 3759 y(maximal)d(ideal)h Fu(f)p Fy(f)38 b Fu(2)28 b Fy(A)1375 3774 y Fx(v)1441 3759 y Fu(j)d Fy(v)t Fz(\()p Fy(f)11 b Fz(\))27 b Fy(>)h Fz(0)p Fu(g)c Fz(of)h Fy(A)2111 3774 y Fx(v)2152 3759 y Fz(;)j(b)m(y)e Fy(F)2398 3774 y Fx(v)2464 3759 y Fz(the)f(residue)i(\014eld)f Fy(A)3225 3774 y Fx(v)3266 3759 y Fy(=3D)-5 b(M)3404 3774 y Fx(v)456 3875 y Fz(of)28 b Fy(v)t= Fz(;)i(b)m(y)f(\000)g(the)g(v)-5 b(aluation)29 b(group)g Fy(v)t Fz(\()p Fy(F)1917 3834 y Fw(\002)1903 3899 y Fl(1)1975 3875 y Fz(\))g(of)f= Fy(v)t Fz(;)i(and)f(b)m(y)g Fy(U)40 b Fz(the)29 b(group)g Fy(A)3190 3890 y Fx(v)3245 3875 y Fu(n)14 b Fy(M)3403 3890 y Fx(v)456 3991 y Fz(of)32 b(units)h(of)f Fy(v)t Fz(.)555 4190= y(The)e(follo)m (wing)e(lemma)i(is)f(w)m(ell)g(kno)m(wn)h(and)f(w)m(e)g(shall)g(omit)g (its)g(straigh)m(tfor-)456 4306 y(w)m(ard)k(pro)s(of.)456 4479 y FF(Lemma)43 b(1.)g Fp(L)-5 b(et)39 b Fy(F)1222 4494 y Fl(1)1300 4479 y Fp(b)-5 b(e)38 b(a)g(value)-5 b(d)38 b(\014eld)f(with)h(valuation)g Fy(v)t= Fp(,)h(valuation)f(gr)-5 b(oup)456 4596 y Fz(\000)31 b(=3D)g Fy(v)t Fz(\()p Fy(F)821 4554 y Fw(\002)807 4620 y Fl(1)879 4596 y Fz(\))p Fp(,)37 b(and)g(gr)-5 b(oup)36 b(of)h(units)g Fy(U)10 b Fp(.)51 b(F)-7 b(or)36 b(e)-5 b(ach)36 b(prime)g Fy(p)31 b Fu(6)p Fz(=3D)h(c)m(har\()p Fy(F)3122 4611 y Fl(1)3162 4596 y Fz(\))k Fp(ther)-5 b(e)456 4712 y(exists)34 b(an)h(isomorphism)1289 4869 y Fy(')27 b Fz(:)h Fy(F)1512 4828 y Fw(\002)1498 4893 y Fl(1)1571 4869 y Fy(=3DF)1697 4822 y Fw(\002)p Fx(p)1683 4893 y Fl(1)1818 4869 y Fu(\000)-16 b(!)28 b Fy(U)5 b(=3DU)2203 4828 y Fx(p)2265 4869 y Fu(\010)23 b Fz(\000)p Fy(=3Dp)p Fz(\000)p Fy(:)456 5026 y Fp(In)34 b(p)-5 b(articular)1015 5183 y Fz(dim)1177 5198 y Ft(F)1218 5206 y Fs(p)1275 5183 y Fy(F)1352 5141 y Fw(\002)1338 5207 y Fl(1)1411 5183 y Fy(=3DF)1537 5135 y Fw(\002)p Fx(p)1523 5207 y Fl(1)1658 5183 y Fz(=3D)28 b(dim)1924 5198 y Ft(F)1965 5206 y Fs(p)2022 5183 y Fy(U)5 b(=3DU)2218 5141 y Fx(p)2280 5183 y Fz(+)22 b(dim)2541 5198 y Ft(F)2582 5206 y Fs(p)2638 5183 y Fz(\000)p Fy(=3Dp)p Fz(\000)p Fy(:)p eop %%Page: 6 6 6 5 bop 456 255 a Fq(6)797 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y= Fz(It)38 b(is)g(w)m(ell-kno)m(wn)i(that)e(for)f(eac)m(h)i(\014eld)g Fy(F)2157 465 y Fl(0)2234 450 y= Fz(and)f(for)f(eac)m(h)i(totally)f (ordered)456 566 y(ab)s(elian)45= b(group)h(\000,)j(there)d(exists)i(a)d (\014eld)i Fy(F)2161 581 y Fl(1)2246 566 y Fz(with)f(a)g(v)-5 b(aluation)45 b Fy(v)54 b Fz(:)c Fy(F)3255 581 y Fl(1)3345 566 y Fu(!)456 683 y Fz(\000)29 b Fu([)g(f1g)42 b= Fz(suc)m(h)i(that)f (the)g(residue)h(\014eld)g Fy(F)2139 698 y Fx(v)2222 683 y Fz(is)g(isomorphic)f(to)g Fy(F)3027 698 y Fl(0)3109 683 y Fz(and)g(the)456 799 y(v)-5 b(aluation)32 b(group)g(is)i(\000.) 555 1004 y(In)f(order)g(to)f(construct)i(suc)m(h)g(a)e(\014eld,)i(set) 757 1175 y Fy(F)820 1190 y Fl(1)887 1175 y Fz(=3D)27 b Fy(F)1053 1190 y Fl(0)1093 1175 y Fz(\(\(\000\)\))g(:=3D)h Fu(f)p Fy(f)38 b Fz(:)28 b(\000)g Fu(!)f Fy(F)1934 1190 y Fl(0)2006 1175 y Fu(j)49 b Fz(supp)q(\()p Fy(f)11 b Fz(\))32 b(is)h(w)m(ell-ordered)r Fu(g)p Fy(:)456 1346 y Fz(Th)m(us)42 b(a)e(t)m(ypical)i(elemen)m(t)g Fy(f)52 b Fu(2)41 b Fy(F)1766 1361 y Fl(1)1846 1346 y= Fz(can)g(b)s(e)g(written) g(as)g(a)f(formal)g(sum)i Fy(f)52 b Fz(=3D)456 1388 y Fo(P)561 1491 y Fx(g)r Fw(2)p Fl(\000)709 1462 y Fy(a)760 1477 y Fx(g)800 1462 y Fy(t)835 1426 y Fx(g)916 1462 y= Fz(suc)m(h)42 b(that)f(the)h(set)f(supp)q(\()p Fy(f)11 b Fz(\))42 b(:=3D)g Fu(f)p Fy(g)j Fu(2)d Fy(G)f Fu(j)f Fy(a)2711 1477 y Fx(g)2793 1462 y Fu(6)p Fz(=3D)i(0)p Fu(g)f Fz(is)g(a)g(w)m(ell-)456 1587 y(ordered)29 b(subset)h(of)f Fy(G)p Fz(.)42 b(The)30 b(v)-5 b(aluation)28 b Fy(v)33 b Fz(on)28 b Fy(f)40 b Fz(is)29 b(de\014ned)h(as:)42 b Fy(v)t Fz(\(0\))27 b(=3D)h Fu(1)g Fz(and)456 1704 y Fy(v)t Fz(\()p Fy(f)11 b Fz(\))36 b(=3D)h(min)18 b(supp)q(\()p Fy(f)11 b Fz(\))37 b(for)h Fy(f)48 b Fu(6)p Fz(=3D)37 b(0.)61 b(An)38= b(imp)s(ortan)m(t)g(prop)s (ert)m(y)h(of)f(the)h(v)-5 b(alued)456 1820 y(\014eld)31 b Fy(F)728 1835 y Fl(1)798 1820 y= Fz(as)f(ab)s(o)m(v)m(e)h(is)g(the)g (fact)f(that)g(it)g(is)h(henselian.)44 b(\(See)32= b(for)d(example)j ([Rib,)456 1936 y(\(1.3\)].\))40 b(In)24= b(what)g(follo)m(ws)h(w)m(e)g (will)g(iden)m(tify)g Fy(F)2169 1951 y Fx(v)2234 1936 y Fz(with)g Fy(F)2511 1951 y Fl(0)2550 1936 y Fz(.)41 b(W)-8 b(e)24 b(will)h(also)f(assume)456 2052 y(that)32 b(c)m(har)h Fy(F)944 2067 y Fl(0)1011 2052 y Fu(6)p Fz(=3D)28 b Fy(p)p Fz(.)555 2257 y(W)-8 b(e)25= b(will)g(b)s(e)f(particularly)h (in)m(terested)i(in)d(con)m(trolling)h(the)g Fy(p)p= Fz(th-p)s(o)m(w)m (er)f(classes)456 2374 y(of)e(suc)m(h)j(a)d(\014eld.)41 b(T)-8 b(o)24= b(do)e(so,)k(w)m(e)e(c)m(ho)s(ose)g(particular)f(groups)g (\000)g(for)g(our)f(v)-5 b(aluation)456 2490 y(groups.)43 b(These)35 b(groups)e(will)g(b)s(e)g(direct)g(sums)h(of)592 2687 y Fv(Z)658 2702 y Fl(\()p Fx(p)p Fl(\))781 2687 y Fz(:=3D)912 2576 y Fo(n)988 2620 y Fy(a)p 988 2664 52 4 v 993 2755 a(b)1077 2687 y Fu(2)28 b Fv(Q)1281 2602 y Fo(\014)1281 2662 y(\014)1347 2687 y Fy(a;)17 b(b)28 b Fu(2)g Fv(Z)p Fy(;)17 b(b)28 b Fu(6)p Fz(=3D)g(0;)49 b(if)32 b Fy(a)c Fu(6)p Fz(=3D)g(0)k(then)h(\()p Fy(a;)17 b(b)p Fz(\))28 b(=3D)g(1)p Fy(;)17 b(p)27 b Fv(-)g Fy(b)3197 2576 y Fo(o)3280 2687 y Fy(:)456 2888 y Fz(Observ)m(e)33 b(that)e Fv(Z)1105 2903 y Fl(\()p Fx(p)p Fl(\))1232 2888 y Fz(is)h(the)g(v)-5 b(aluation)31 b(ring)g(of)g(a)h Fy(p)p Fz(-adic)f(v)-5 b(aluation)31 b(on)h Fv(Q)p Fz(.)43 b(Let)456 3004 y Fy(I)d Fz(b)s(e)33 b(an)m(y)g(non-empt)m(y)-8 b(,)34 b(w)m(ell-ordered)g(set.)44 b(Then)34 b(set)986 3201 y(\000)28 b(=3D)f Fv(Z)1244 3150 y Fl(\()p Fx(I)5 b Fl(\))1244 3232 y(\()p Fx(p)p Fl(\))1367 3201 y Fz(:=3D)1498 3090 y Fo(n)1564 3201 y Fy(\015)33 b Fz(:)28 b Fy(I)35 b Fu(!)28 b Fv(Z)1975 3217 y Fl(\()p Fx(p)p Fl(\))2102 3086 y Fo(\014)2102 3146 y(\014)2102 3206 y(\014)2168 3201 y Fu(j)17 b Fz(supp\()p Fy(\015)5 b Fz(\))p Fu(j)27 b Fy(<)h Fu(1)2804 3090 y Fo(o)2887 3201 y Fy(:)456 3404 y Fz(Th)m(us)37 b(\000)e(is)i(a)e(direct)h(sum)h(of)e Fu(j)p Fy(I)8 b Fu(j)35 b Fz(copies)i(of)e Fv(Z)2204 3419 y Fl(\()p Fx(p)p Fl(\))2299 3404 y Fz(.)53 b(Observ)m(e)37 b(that)f Fv(Z)3037 3419 y Fl(\()p Fx(p)p Fl(\))3167 3404 y Fz(carries)456 3520 y(a)h(natural)g(ordering)h(induced)h(from)f Fv(Q)p Fz(,)h(and)f(then)g(w)m(e)h(ma)m(y)f(order)g(\000)g(lexico-)456 3636 y(graphically)-8 b(,)38 b(as)f(follo)m(ws.)57 b(Let)37 b Fy(\015)1706 3651 y Fl(1)1780 3636 y Fu(6)p Fz(=3D)d Fy(\015)1941 3651 y Fl(2)2015 3636 y Fu(2)i Fz(\000.)56 b(Then)37 b Fy(\015)2570 3651 y Fl(1)2644 3636 y Fy(<)e(\015)2806 3651 y Fl(2)2882 3636 y Fz(if)h(and)h(only)g(if)456 3752 y Fy(\015)507 3767 y Fl(1)546 3752 y Fz(\()p Fy(i)p Fz(\))30 b Fy(<)g(\015)842 3767 y Fl(2)881 3752 y Fz(\()p= Fy(i)p Fz(\))k(for)f(the)i(least)f(elemen)m(t)i Fy(i)30 b Fu(2)h Fy(I)41 b Fz(suc)m(h)36 b(that)d Fy(\015)2664 3767 y Fl(1)2703 3752 y Fz(\()p Fy(i)p Fz(\))e Fu(6)p Fz(=3D)e Fy(\015)2999 3767 y Fl(2)3038 3752 y Fz(\()p Fy(i)p Fz(\).)48 b(Then)456 3869 y(\000)34= b(is)i(a)e(linearly)i(ordered)f(ab)s(elian)g (group.)50 b(Recall)35 b(that)g(eac)m(h)h(non-empt)m(y)g(set)456 3985 y(can)c(b)s(e)h(w)m(ell-ordered.)45 b(\(See)34= b([La)o(,)f(App)s (endix)h(2,)e(Theorem)i(4.1].\))555 4190 y(W)-8= b(e)37 b(c)m(ho)s(ose)h(\000)f(as)g(ab)s(o)m(v)m(e)g(b)s(ecause)h Fy(G)1985 4205 y Fx(F)2030 4214 y Fm(0)2064 4205 y Fl(\(\(\000\)\))2259 4190 y Fz(will)f(b)s(e)g(pro-)p Fy(p)f Fz(\(see)i(Lemma)g(3)456 4306 y(b)s(elo)m(w\))48= b(and)f(b)s(ecause)i(w)m(e)f(ma)m(y)g(con)m (trol)f(the)h Fy(p)p Fz(th-p)s(o)m(w)m(er)g(classes)h(with)f(the)456 4422 y(follo)m(wing)36 b(lemma.)57 b(This)37= b(w)m(ell-kno)m(wn)i (lemma)e(follo)m(ws)h(from)e(Lemma)h(1)f(and)456= 4539 y(the)g(fact)h(that)f(the)h(v)-5 b(alued)37 b(\014eld)g Fy(F)1799 4554 y Fl(1)1875 4539 y Fz(is)g(henselian.)57 b(It)36 b(is)h(also)g(an)f(immediate)456 4655= y(consequence)f(of)d([W)q (,)g(Lemma)h(1.4].)44 b(Therefore)33= b(w)m(e)h(shall)f(omit)g(its)g (pro)s(of.)456 4837 y FF(Lemma)39 b(2.)i Fp(L)-5 b(et)36 b Fy(F)1213 4852 y Fl(1)1280 4837 y Fz(=3D)27 b Fy(F)1446 4852 y Fl(0)1486 4837 y Fz(\(\(\000\)\))35 b Fp(as)f(ab)-5 b(ove.)44 b(Then)969 5013 y Fz(dim)1131 5028 y Ft(F)1172 5036 y Fs(p)1229 5013 y Fy(F)1306 4972 y Fw(\002)1292 5037 y Fl(1)1364 5013 y Fy(=3DF)1490 4966 y Fw(\002)p Fx(p)1476 5037 y Fl(1)1612 5013 y Fz(=3D)28 b(dim)1878 5028 y Ft(F)1919 5036 y Fs(p)1976 5013 y Fy(F)2053 4972 y Fw(\002)2039 5037 y Fl(0)2111 5013 y Fy(=3DF)2237 4966 y Fw(\002)p Fx(p)2223 5037 y Fl(0)2354 5013 y Fz(+)22 b(dim)2614 5028 y Ft(F)2655 5036 y Fs(p)2712 5013 y Fz(\000)p Fy(=3Dp)p Fz(\000)1612 5177 y(=3D)28 b(dim)1878 5192 y Ft(F)1919 5200 y Fs(p)1976 5177 y Fy(F)2053 5135 y Fw(\002)2039 5201 y Fl(0)2111 5177 y Fy(=3DF)2237 5129 y Fw(\002)p Fx(p)2223 5201 y Fl(0)2354 5177 y Fz(+)22 b Fu(j)p Fy(I)8 b Fu(j)p Fy(:)p eop %%Page: 7 7 7 6 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)330 b(7)555 450 y Fz(Finally)-8 b(,)33 b(w)m(e)h(record)f(a)f(criterion)h(for)f Fy(G)2045 465 y Fx(F)2090 474 y Fm(1)2161 450 y Fz(b)s(eing)h(pro-)p Fy(p)p Fz(:)456 609 y FF(Lemma)53 b(3.)48 b Fp(L)-5 b(et)47 b Fy(F)1245 624 y Fl(1)1333 609 y Fz(=3D)i Fy(F)1521 624 y Fl(0)1561 609 y Fz(\(\(\000\)\))d Fp(as)g(ab)-5 b(ove)45 b(with)h Fz(c)m(har)q(\()p Fy(F)2737 624 y Fl(0)2777 609 y Fz(\))i(=3D)h(0)d Fp(and)g Fy(G)3361 624 y Fx(F)3406 633 y Fm(0)456 725 y Fp(pr)-5 b(o-)p Fy(p)p Fp(.)44 b(Then)34 b Fy(G)1081 740 y Fx(F)1126 749 y Fm(1)1199 725 y Fp(is)h(pr)-5 b(o-)p Fy(p)34 b Fp(as)h(wel)5 b(l.)456 963 y(Pr)-5 b(o)g(of.)41 b Fz(F)-8 b(rom)35 b(basic)h(v)-5 b(aluation)35 b(theory)-8 b(,)37 b(nicely)g(summarized)g(in)f([K,)g(pages)g(3)456 1079 y(and)c(4],)h(and)f(the)h(fact)g(that)f Fy(F)1605 1094 y Fl(1)1677 1079 y Fz(ab)s(o)m(v)m(e)i(is)f(henselian,)1613 1236 y Fy(G)1690 1251 y Fx(F)1735 1260 y Fm(1)1801 1209 y Fu(\030)1802 1240 y Fz(=3D)1907 1236 y Fy(T)i Fv(o)23 b Fy(G)2176 1251 y Fx(F)2221 1260 y Fm(0)2259 1236 y Fy(;)456 1393 y Fz(where)29 b(the)f(action)g(of)g Fy(G)1368 1408 y Fx(F)1413 1417 y Fm(0)1479 1393 y Fz(on)g Fy(T)41 b Fz(is)29 b(uniquely)h(determined)g(b)m(y)e(the)h(cyclotomic)456 1509 y(c)m(haracter)40 b(mapping)f Fy(G)1372 1524 y Fx(F)1417 1533 y Fm(0)1494 1509 y= Fz(in)m(to)h(a)f(group)f(of)h(automorphisms)i (of)d(a)h(group)g(of)456 1626 y(ro)s(ots)30= b(of)g(unit)m(y)i(con)m (tained)g(in)f Fy(F)1692 1589 y Fx(s)1678 1650 y Fl(0)1728 1626 y Fz(,)g(and)g Fy(T)2072 1598 y Fu(\030)2073 1630 y Fz(=3D)2177 1626 y Fv(Z)2243 1589 y Fx(I)2243 1650 y(p)2284 1626 y Fz(,)g(the)g(top)s(ological)f(pro)s(duct)h(of)456 1746 y Fu(j)p Fy(I)8 b Fu(j)31 b Fz(copies)j(of)e Fv(Z)1059 1761 y Fx(p)1099 1746 y Fz(.)44 b(In)33 b(particular,)g(if)f Fy(G)1935 1761 y Fx(F)1980 1770 y Fm(0)2051 1746 y Fz(is)h(a)f(pro-)p Fy(p)p Fz(-group,)g(so)g(is)h Fy(G)3083 1761 y Fx(F)3128 1770 y Fm(1)3167 1746 y Fz(.)173 b Fi(\003)456 1992 y Fz(2.2.)48 b Fy(H)745 1956 y Fl(2)784 1992 y Fz(\()p Fy(G)899 2007 y Fx(F)944 2016 y Fm(1)982 1992 y Fy(;)17 b Fv(F)1087 2007 y Fx(p)1127 1992 y Fz(\))37 b FF(for)g(henselian)j(v) -6 b(alued)38 b(\014elds)h Fy(F)2566 2007 y Fl(1)2605 1992 y FF(.)555 2181 y Fz(No)m(w)g(w)m(e)h(study)g Fy(H)1297 2144 y Fl(2)1335 2181 y Fz(\()p Fy(G)1450 2196 y Fx(F)1495 2205 y Fm(1)1534 2181 y Fy(;)17 b Fv(F)1639 2196 y Fx(p)1678 2181 y Fz(\))38 b(for)g(our)h(henselian)h(v)-5 b(alued)39 b(\014elds)h Fy(F)3149 2196 y Fl(1)3188 2181 y Fz(.)61 b(The)456 2297 y(next)29= b(lemma,)i(tak)m(en)f(from)f([W],)h(will)f(b)s (e)h(used)g(in)f(the)g(pro)s(of)f(of)h(Theorem)h(1)f(to)456 2413 y(sho)m(w)k(that)g(norm)f(groups)h(of)f(cyclic)j Fy(p)p Fz(-extensions)f(of)e Fy(F)2583 2428 y Fl(1)2655 2413 y Fz(are)h(not)f(to)s(o)g(large.)555 2601 y(Supp)s(ose)49 b(that)f Fy(F)1243 2616 y Fl(1)1331 2601 y= Fz(is)h(a)e(\014eld)i(endo)m (w)m(ed)h(with)f(a)f(henselian)i(v)-5 b(aluation)48 b Fy(v)456 2717 y Fz(with)43 b(v)-5 b(aluation)44 b(group)f(\000)i(=3D)h Fy(v)t Fz(\()p Fy(F)1806 2676 y Fw(\002)1792 2742 y Fl(1)1864 2717 y Fz(\).)76 b(Let)43 b Fy(F)2267 2681 y Fx(nr)2253 2742 y Fl(1)2391 2717 y Fz(denote)h(the)g(maximal)f(un-)456 2834 y(rami\014ed)49 b(extension)i(of)d Fy(F)1489 2849 y Fl(1)1578 2834 y Fz(in)h(its)g(separable)h(closure)g Fy(F)2724 2797 y Fx(s)2710 2858 y Fl(1)2761 2834 y Fz(.)92 b(Then)50 b Fy(G)3228 2849 y Fx(F)3273 2858 y Fm(0)3367 2806 y Fu(\030)3368 2838 y Fz(=3D)456 2950 y Fy(G)533 2965 y Fx(F)578 2974 y Fm(1)616 2950 y Fy(=3D)17 b Fz(Gal)o(\()p Fy(F)949 2914 y Fx(s)935 2974 y Fl(1)985 2950 y Fy(=3DF)1111 2914 y Fx(nr)1097 2974 y Fl(1)1191 2950 y Fz(\).)54= b(Therefore,)38 b(after)e(iden)m(tifying)h(these)h(groups,)f(w)m(e)g(ha)m(v)m(e)456 3066 y(the)c(in\015ation)f(map)1045 3223 y(inf)j(=3D)27 b(inf)1429 3180 y Fx(F)1474 3189 y Fm(1)1429 3250 y Fx(F)1474 3259 y Fm(0)1540 3223 y Fz(:)h Fy(H)1684 3182 y Fw(\003)1723 3223 y Fz(\()p Fy(G)1838 3238 y Fx(F)1883 3247 y Fm(0)1921 3223 y Fy(;)17 b Fv(F)2026 3238 y Fx(p)2065 3223 y Fz(\))28 b Fu(\000)-16 b(!)27 b Fy(H)2408 3182 y Fw(\003)2447 3223 y Fz(\()p Fy(G)2562 3238 y Fx(F)2607 3247 y Fm(1)2646 3223 y Fy(;)17 b Fv(F)2751 3238 y Fx(p)2790 3223 y Fz(\))p Fy(:)456 3383 y Fz(\(See)33 b([W,)g(page)g(483].\))555 3571= y(Moreo)m(v)m(er,)e(from)e(basic)h(Kummer)f(theory)h(w)m(e)g(ha)m (v)m(e)g(the)f(canonical)h(isomor-)456 3688 y(phism)1306 3804 y Fy(')1370 3819 y Fx(F)1457 3804 y Fz(:)d Fy(F)1588 3763 y Fw(\002)1647 3804 y Fy(=3DF)1773 3763 y Fw(\002)p Fx(p)1895 3804 y Fu(\000)-16 b(!)27 b Fy(H)2172 3763 y Fl(1)2211 3804 y Fz(\()p Fy(G)2326 3819 y Fx(F)2384 3804 y Fy(;)17 b Fv(F)2489 3819 y Fx(p)2529 3804 y Fz(\))p Fy(;)456 3940 y Fz(as)32= b(w)m(ell)h(as)f(the)g(corresp)s(onding)h (canonical)g(isomorphisms)g Fy(')2783 3955 y Fx(F)2828 3965 y Fs(i)2859 3940 y Fz(,)f Fy(i)c Fz(=3D)f(1,)32 b(2.)43 b(W)-8 b(e)456 4057 y(will)44 b(denote)g(b)m(y)g(\()p Fy(f)11 b Fz(\))1257 4072 y Fx(F)1358 4057 y Fz(or)43 b(\()p Fy(f)1574 4072 y Fx(i)1602 4057 y Fz(\))1640 4072 y Fx(F)1685 4082 y Fs(i)1759 4057 y Fz(the)g(images)h Fy(')2332 4072 y Fx(F)2391 4057 y Fz(\([)p Fy(f)11 b Fz(]\))43 b(or)g Fy(')2817 4072 y Fx(F)2862 4082 y Fs(i)2892 4057 y Fz(\([)p Fy(f)3005 4072 y Fx(i)3033 4057 y Fz(]\).)76 b(If)43 b(the)456 4173 y(con)m(text)33= b(is)h(clear)f(w)m(e)g(will)g (omit)g(the)g(subscript.)555 4361 y(Assume)j(next)f(that)f Fu(f)p Fy(\031)1457 4376 y Fx(j)1493 4361 y Fy(;)51 b(j)36 b Fu(2)31 b(J)18 b(g)33 b= Fz(is)i(a)e(set)i(of)f(elemen)m(ts)i(of)d Fy(F)2951 4320 y Fw(\002)2937 4385 y Fl(1)3044 4361 y Fz(suc)m(h)j(that)456 4477 y(their)e(images)g(in)g(\000)p Fy(=3Dp)p Fz(\000)f(form)g(a)h(basis)g(of)g(\000)p= Fy(=3Dp)p Fz(\000)f(o)m(v)m(er)h Fv(F)2569 4492 y Fx(p)2609 4477 y Fz(.)47 b(Then)34 b(w)m(e)h(ha)m(v)m(e)g(the)456 4594 y(follo)m(wing)25= b(lemma,)i(obtained)f(as)f(a)g(sp)s(ecial)h(case)g (of)f(a)g(theorem)h(of)e(W)-8 b(adsw)m(orth.)456 4752 y FF(Lemma)44 b(4.)g Fz([W)q(,)38 b(Theorem)h(3.6,)f(page)f(483])p Fp(.)58 b(L)-5 b(et)39 b Fy(F)2518 4767 y Fl(1)2597 4752 y Fp(and)g Fy(F)2854 4767 y Fl(0)2933 4752 y Fp(b)-5 b(e)38 b(as)h(ab)-5 b(ove.)456 4869 y(Then)616 5025 y Fy(H)705 4984 y Fl(2)744 5025 y Fz(\()p Fy(G)859 5040 y Fx(F)904 5049 y Fm(1)942 5025 y Fy(;)17 b Fv(F)1047 5040 y Fx(p)1086 5025 y Fz(\))28 b(=3D)16 b(inf)7 b(\()p Fy(H)1489 4984 y Fl(2)1528 5025 y Fz(\()p Fy(G)1643 5040 y Fx(F)1688 5049 y Fm(0)1727 5025 y Fy(;)17 b Fv(F)1832 5040 y Fx(p)1871 5025 y Fz(\)\))22 b Fu(\010)2046 5040 y Fx(j)t Fw(2J)2212 4945 y Fo(\000)2258 5025 y Fz(inf)7 b(\()p Fy(H)2503 4984 y Fl(1)2542 5025 y Fz(\()p Fy(G)2657 5040 y Fx(F)2702 5049 y Fm(0)2740 5025 y Fy(;)17 b Fv(F)2845 5040 y Fx(p)2884 5025 y Fz(\)\))22 b Fu([)h Fz(\()p Fy(\031)3164 5040 y Fx(j)3201 5025 y Fz(\))3239 4945 y Fo(\001)1250 5177 y Fu(\010)1327 5192 y Fw(f)p Fx(j)1391 5201 y Fm(1)1426 5192 y Fx(;j)1475 5201 y Fm(2)1508 5192 y Fw(g\032J)13 b Fx(;)34 b(j)1742 5201 y Fm(1)1776 5192 y Fw(6)p Fl(=3D)p Fx(j)1860 5201 y Fm(2)1921 5177 y Fz(\(\()p Fy(\031)2052 5192 y Fx(j)2081 5201 y Fm(1)2119 5177 y Fz(\))22 b Fu([)h Fz(\()p Fy(\031)2361 5192 y Fx(j)2390 5201 y Fm(2)2428 5177 y Fz(\)\))17 b Fy(:)p eop %%Page: 8 8 8 7 bop 456 255 a Fq(8)797 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456 450 y= Fp(Mor)k(e)g(over,)34 b(for)h(e)-5 b(ach)34 b Fy(j)g Fu(2)28 b(J)18 b Fp(,)1042 619 y Fz(inf)7 b(\()p Fy(H)1287 578 y Fl(1)1326 619 y Fz(\()p Fy(G)1441 634 y Fx(F)1486 643 y Fm(0)1524 619 y Fy(;)17 b Fv(F)1629 634 y Fx(p)1668 619 y Fz(\)\))1772 591 y Fu(\030)1773 623 y Fz(=3D)1877 619 y(inf)7 b(\()p Fy(H)2122 578 y Fl(1)2161 619 y Fz(\()p Fy(G)2276 634 y Fx(F)2321 643 y Fm(0)2359 619 y Fy(;)17 b Fv(F)2464 634 y Fx(p)2504 619 y Fz(\)\))22 b Fu([)g Fz(\()p Fy(\031)2783 634 y Fx(j)2820 619 y Fz(\))456 788 y Fp(and)34 b(for)h(e)-5 b(ach)34 b Fy(j)1060 803 y Fl(1)1100 788 y Fy(;)17 b(j)1184 803 y Fl(2)1251 788 y Fu(2)28 b(J)53 b Fp(such)34 b(that)i Fy(j)1927 803 y Fl(1)1994 788 y Fu(6)p Fz(=3D)28 b Fy(j)2138 803 y Fl(2)2177 788 y Fp(,)35 b(we)g(have)f Fz(\()p Fy(\031)2704 803 y Fx(j)2733 812 y Fm(1)2771 788 y Fz(\))22 b Fu([)h Fz(\()p Fy(\031)3013 803 y Fx(j)3042 812 y Fm(2)3080 788 y Fz(\))28 b Fu(6)p Fz(=3D)g(0)p Fp(.)555 1056 y= Fz(Note)44 b(that)g(in)g(the)g(last)g(summand)i(of)d(Lemma)h(4)g(the)g(sum)h (ranges)f(o)m(v)m(er)456 1173 y(subsets)f Fu(f)p Fy(j)892 1188 y Fl(1)931 1173 y Fy(;)17 b(j)1015 1188 y Fl(2)1055 1173 y Fu(g)p Fy(;)g(j)1189 1188 y Fl(1)1270 1173 y Fu(6)p Fz(=3D)42 b Fy(j)1428 1188 y Fl(2)1509 1173 y Fz(of)f Fu(J)59 b Fz(and)41 b(a)f(c)m(hoice)j(b)s(et)m(w)m(een)g(\()p Fy(\031)2820 1188 y Fx(j)2849 1197 y Fm(1)2888 1173 y Fz(\))27 b Fu([)i Fz(\()p Fy(\031)3141 1188 y Fx(j)3170 1197 y Fm(2)3208 1173 y Fz(\))41 b(and)456 1289 y(\()p Fy(\031)549 1304 y Fx(j)578 1313 y Fm(2)616 1289 y Fz(\))22 b Fu([)h Fz(\()p Fy(\031)858 1304 y Fx(j)887 1313 y Fm(1)925 1289 y Fz(\))33 b(is)g(arbitrary)f(but)h(\014xed.)456 1575 y(2.3.)48 b FF(Residue)37 b(\014elds)f Fy(F)1427 1590 y Fl(0)1502 1575 y= FF(with)g(prescrib)s(ed)h(absolute)f(Galois)h (group.)555 1779 y Fz(In)f(our)f(construction)h(of)f Fy(F)49 b Fz(w)m(e)36 b(c)m(ho)s(ose)g(a)f(residue)h(\014eld)g Fy(F)2795 1794 y Fl(0)2870 1779 y Fz(dep)s(ending)g(on)456 1895 y(\007)j(and)g Fy(p)p Fz(.)63 b(If)39= b(\007)g(=3D)g(1)g(w)m(e)h (will)g(simply)h(put)e Fy(F)2236 1910 y Fl(0)2314 1895 y Fz(=3D)g Fv(C)p Fz(,)i(but)f(when)g(\007)f(=3D)f(0)h(w)m(e)456 2011 y(require)d(some)g(sp)s(ecial)h(prop)s(erties)f(of)f Fy(F)1992 2026 y Fl(0)2031 2011 y Fz(.)52 b(In)36= b(particular,)g(in)g (order)f(that)g(our)456 2128 y(cyclic)46 b(extension)h Fy(K)55 b Fz(=3D)49 b Fy(F)14 b Fz(\()1585 2094 y Fs(p)1562 2056 y Fu(p)p 1645 2056 52 4 v 72 x Fy(a)p Fz(\))45= b(ha)m(v)m(e)h(the) f(desired)h(in)m(v)-5 b(arian)m(t)46 b(\007)i(=3D)h(0,)e(w)m(e)456 2244 y(require)37 b(that)f Fy(K)43 b Fz(do)s(es)37= b(not)f(em)m(b)s(ed) h(in)g(a)e(cyclic)j(Galois)e(extension)i Fy(L)e Fz(o)m(v)m(er)i Fy(F)456 2360 y Fz(with)f(degree)h([)p Fy(L)e Fz(:)f Fy(F)14 b Fz(])36 b(=3D)f Fy(p)1481 2324 y Fl(2)1520 2360 y Fz(,)j(for)f(this)h(is)f(equiv)-5 b(alen)m(t)39 b(to)e Fy(\030)2671 2375 y Fx(p)2757 2360 y Fy(=3D)-61 b Fu(2)36 b Fy(N)2925 2375 y Fx(K)q(=3DF)3079 2360 y Fz(\()p Fy(K)3207 2324 y Fw(\002)3266 2360 y Fz(\))h(b)m(y)456 2476= y([A,)32 b(Theorem)i(3].)555 2680 y(T)-8 b(o)47= b(ensure)i(that)d(this)i(nonem)m (b)s(eddabilit)m(y)i(condition)d(holds,)52 b(as)47 b(w)m(ell)h(as)456 2797 y(to)37= b(ensure)j(that)d(a)h(certain)h(nonab)s(elian)f(group)g (of)f(order)h Fy(p)2732 2760 y Fl(3)2810 2797 y= Fz(do)s(es)g(not)g(o)s (ccur)456 2913= y(as)g(a)f(Galois)h(group)g(o)m(v)m(er)g(the)h(\014eld,) h(w)m(e)f(c)m(ho)s(ose)f(residue)i(\014elds)f Fy(F)3004 2928 y Fl(0)3081 2913 y Fz(with)g(ab-)456 3029= y(solute)j(Galois)g (groups)g(taking)g(a)g(sp)s(ecial)g(form,)i(and)e(it)g(is)h(also)f(con) m(v)m(enien)m(t)456 3145 y(to)h(require)j(that)e Fu(j)p Fy(F)1254 3104 y Fw(\002)1240 3170 y Fl(0)1312 3145 y Fy(=3DF)1438 3098 y Fw(\002)p Fx(p)1424 3170 y Fl(0)1532 3145 y Fu(j)g Fz(is)h(small.)79 b(As)45= b(it)f(turns)h(out,)i(w)m(e)e (ma)m(y)g(c)m(ho)s(ose)456 3261 y(some)29= b(suitable)g(algebraic)g (in\014nite)h(extension)g(of)e Fv(Q)p Fz(.)42= b(Finitely)30 b(generated)f(pro-)456 3378 y Fy(p)p= Fz(-absolute)39 b(Galois)f(groups)i(o)m(v)m(er)g Fv(Q)f= Fz(and)g(more)g(generally)h(an) m(y)g(global)e(\014eld,)456 3494 y(w)m(ere)31= b(nicely)g(classi\014ed)h (in)e([E2)q(].)42 b(\(See)31= b(also)f([E1])g(and)g([JP)q(])g(for)f (related)i(results)456 3610 y(and)h(tec)m(hniques.\))555 3814 y(The)i(extensions)h(w)m(e)e(will)g(need)h(for)e Fy(p)c(>)f Fz(2)33 b(are)f(giv)m(en)i(in)f(the)g(follo)m(wing)456 3995 y FF(Lemma)c(5.)36 b Fz([E2,)26 b(page)f(84])i Fp(F)-7 b(or)26 b(e)-5 b(ach)27 b(prime)g Fy(p)g(>)h Fz(2)f Fp(ther)-5 b(e)27 b(exists)h(an)f(algebr)-5 b(aic)456 4112 y(extension)33 b Fy(F)949 4127 y Fl(0)p Fx(;p)1079 4112 y Fp(of)i= Fv(Q)g Fp(such)g(that)1243 4284 y Fy(G)1320 4299 y Fx(F)1365 4308 y Fm(0)p Fs(;p)1482 4284 y Fz(=3D)28 b Fu(h)p Fy(\033)n(;)17 b(\034)46 b Fu(j)35 b Fy(\033)t(\034)11 b(\033)2044 4243 y Fw(\000)p Fl(1)2166 4284 y Fz(=3D)28 b Fy(\034)2323 4243 y Fx(p)p Fl(+1)2453 4284 y Fu(i)2492 4299 y Fh(pr)l(o-)p Fx(p)456 4453 y Fp(wher)-5 b(e)34 b(the)h(pr)-5= b(esentation)34 b(is)h(in)f(the)h(c)-5 b(ate)g(gory)35 b(of)f(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)555 4722 y Fz(Observ)m(e)31= b(that)d(the)h (maximal)g(ab)s(elian)g(extension)h Fy(F)2530 4686 y Fx(ab)2516 4746 y Fl(0)p Fx(;p)2639 4722 y Fz(of)e Fy(F)2809 4737 y Fl(0)p Fx(;p)2933 4722 y Fz(has)g Fy(G)3179 4686 y Fx(ab)3179 4747 y(F)3224 4756 y Fm(0)p Fs(;p)3341 4722 y Fz(:=3D)456 4856 y(Gal)o(\()p Fy(F)723 4820 y Fx(ab)709 4881 y Fl(0)p Fx(;p)804 4856 y Fy(=3DF)916 4871 y Fl(0)p Fx(;p)1010 4856 y Fz(\))33 b(equal)g(to)1183 5042 y Fy(G)1260 5001 y Fx(ab)1260 5067 y(F)1305 5076 y Fm(0)p Fs(;p)1422 5014 y Fu(\030)1423 5046 y Fz(=3D)1527 5042 y Fu(h)5 b Fz(\026)-54 b Fy(\033)t(;)22 b Fz(\026)-54 b Fy(\034)44 b Fu(j)37 b Fz(\026)-54 b Fy(\034)1868 5001 y Fx(p)1936 5042 y Fz(=3D)27 b(1)p Fu(i)h Fz(=3D)f Fv(Z)2324 5057 y Fx(p)2386 5042 y Fu(\002)c Fv(Z)p Fy(=3Dp)p Fv(Z)456 5216 y Fz(for)32 b(all)g(primes)i Fy(p)28 b(>)f Fz(2.)p eop %%Page: 9 9 9 8 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)330 b(9)456 450 y Fz(2.4.)48 b FF(Field)39 b(arithmetic)e(and)i(free)e(pro-)p Fy(p)h FF(pro)s(ducts.)555 650 y Fz(In)32= b(this)g(section)g(w)m(e)g(collect)g (lemmas)h(giving)e(information)g(ab)s(out)g(a)g(\014eld)h Fy(F)456 766 y Fz(deriv)m(ed)40 b(from)e(the)h(structure)h(of)e Fy(G)1831 781 y Fx(F)1890 766 y Fz(,)i(esp)s(ecially)h(when)e Fy(G)2740 781 y Fx(F)2838 766 y Fz(is)g(a)f(free)h(pro-)p Fy(p)456 882 y Fz(pro)s(duct)32 b(of)h(t)m(w)m(o)g(pro-)p Fy(p)f Fz(groups)h Fy(G)1764 897 y Fx(F)1809 906 y Fm(1)1879 882 y Fz(and)g Fy(G)2146 897 y Fx(F)2191 906 y Fm(2)2229 882 y Fz(.)555 1081 y(First)25= b(w)m(e)h(record)g(a)e(lemma)i (detecting)g(the)g(presence)h(of)d(primitiv)m(e)j= Fy(p)p Fz(th)e(ro)s(ots)456 1198 y(of)32 b(unit)m(y)h(in)g(a)g(\014eld)g Fy(F)14 b Fz(,)32 b(based)i(only)f(on)f(the)h(structure)h(of)e Fy(G)2759 1213 y Fx(F)2818 1198 y Fz(.)456 1372 y FF(Lemma)39 b(6.)j Fp(Supp)-5 b(ose)34 b(that)i Fy(p)28 b(>)g Fz(2)34 b Fp(and)h(that)g Fy(F)49 b Fp(is)35= b(a)g(\014eld)f(with)h Fz(c)m(har)q(\()p Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h Fy(p)456 1488 y Fp(and)34 b Fy(G)722 1503 y Fx(F)815 1488 y Fp(pr)-5 b(o-)p Fy(p)p Fp(.)45 b(Then)34 b Fy(\030)1407 1503 y Fx(p)1474 1488 y Fu(2)28 b Fy(F)1645 1452 y Fw(\002)1703 1488 y Fp(.)456 1738 y(Pr)-5 b(o)g(of.)41 b Fz(Because)23 b(c)m(har)q(\()p Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h= Fy(p)p Fz(,)c(there)e(exists)i(a)e(primitiv)m(e)h Fy(p)p= Fz(th)f(ro)s(ot)g Fy(\030)3061 1753 y Fx(p)3122 1738 y Fz(of)f(unit)m(y)456 1854 y(in)34 b Fy(F)648 1818 y Fx(s)684 1854 y Fz(.)48 b(If)34 b Fy(\030)901 1869 y Fx(p)970 1854 y Fu(2)d Fy(F)1144 1818 y Fx(s)1203 1854 y Fu(n)23 b Fy(F)48 b Fz(then)34 b Fy(F)14 b Fz(\()p Fy(\030)1768 1869 y Fx(p)1807 1854 y Fz(\))p Fy(=3DF)47 b Fz(is)35= b(a)f(non)m(trivial)g(Galois)g(extension) i(of)456 1970 y(degree)i([)p Fy(F)14 b Fz(\()p Fy(\030)949 1985 y Fx(p)988 1970 y Fz(\))35 b(:)g Fy(F)14 b Fz(])35 b Fy(<)h(p)p Fz(.)56 b(Therefore)39 b Fy(G)2030 1985 y Fx(F)2125 1970 y Fz(has)f(a)e(non)m(trivial)i(\014nite)g(quotien)m(t) 456 2087 y(of)29 b(order)i(coprime)g(with)f Fy(p)p Fz(.)43 b(This)31 b(con)m(tradicts)h(our)e(assumption)h(that)f Fy(G)3211 2102 y Fx(F)3300 2087 y Fz(is)h(a)456 2203 y(pro-)p Fy(p)p Fz(-group.)42 b(Hence)34 b Fy(\030)1358 2218 y Fx(p)1425 2203 y Fu(2)28 b Fy(F)1596 2167 y Fw(\002)1687 2203 y Fz(as)33 b(asserted.)1192 b Fi(\003)555 2453 y Fz(No)m(w)34 b(supp)s(ose)g(that)f Fy(G)1434 2468 y Fx(F)1521 2453 y Fz(=3D)28 b Fy(G)1702 2468 y Fx(F)1747 2477 y Fm(1)1808 2453 y Fy(?)22 b(G)1956 2468 y Fx(F)2001 2477 y Fm(2)2072 2453 y Fz(for)32 b(pro-)p Fy(p)h Fz(absolute)g(Galois)g(groups)456 2569 y Fy(G)533 2584 y Fx(F)591 2569 y Fz(,)k Fy(G)732 2584 y Fx(F)777 2593 y Fm(1)815 2569 y Fz(,)g(and)f Fy(G)1149 2584 y Fx(F)1194 2593 y Fm(2)1232 2569 y= Fz(,)h(where)g(the)f(free)g (pro)s(duct)g(is)h(tak)m(en)f(in)g(the)h(category)f(of)456 2685 y(pro-)p Fy(p)p Fz(-groups.)61 b(F)-8 b(rom)38= b([N,)j(\(4.3\))d (Satz])h(w)m(e)h(see)f(that)g(the)g(restriction)h(homo-)456 2801 y(morphism)930 2959 y(res)28 b(:)g Fy(H)1221 2917 y Fl(1)1260 2959 y Fz(\()p Fy(G)1375 2974 y Fx(F)1433 2959 y Fy(;)17 b Fv(F)1538 2974 y Fx(p)1578 2959 y Fz(\))27 b Fu(\000)-16 b(!)28 b Fy(H)1921 2917 y Fl(1)1959 2959 y Fz(\()p Fy(G)2074 2974 y Fx(F)2119 2983 y Fm(1)2158 2959 y Fy(;)17 b Fv(F)2263 2974 y Fx(p)2302 2959 y Fz(\))22 b Fu(\010)h Fy(H)2551 2917 y Fl(1)2590 2959 y Fz(\()p Fy(G)2705 2974 y Fx(F)2750 2983 y Fm(2)2788 2959 y Fy(;)17 b Fv(F)2893 2974 y Fx(p)2932 2959 y Fz(\))350 b(\(1\))456 3116 y(is)24 b(an)g(isomorphism.)42 b(No)m(w)25 b(giv)m(en)g(\()p Fy(f)11 b Fz(\))1872 3131 y Fx(F)1954 3116 y Fz(in)24 b Fy(H)2148 3080 y Fl(1)2187 3116 y Fz(\()p Fy(G)2302 3131 y Fx(F)2360 3116 y Fy(;)17 b Fv(F)2465 3131 y Fx(p)2505 3116 y Fz(\),)25 b(w)m(e)g(denote)g(the)f(image)456 3232 y(res\()p Fy(f)11 b Fz(\))710 3247 y Fx(F)801 3232 y Fz(b)m(y)1382 3349 y(res)q(\()p Fy(f)g Fz(\))1637 3364 y Fx(F)1723 3349 y Fz(=3D)28 b(\()p Fy(f)11 b Fz(\))1962 3364 y Fx(G)2017 3375 y Fs(F)2056 3390 y Fm(1)2120 3349 y Fu(\010)22 b Fz(\()p Fy(f)11 b Fz(\))2354 3364 y Fx(G)2409 3375 y Fs(F)2448 3390 y Fm(2)2491 3349 y Fy(:)456 3493 y Fz(This)27 b(notation)f(distinguishes,)31= b(then,)d(b)s(et)m(w)m(een) h(\()p Fy(f)11 b Fz(\))2405 3508 y Fx(F)2450 3517 y Fm(1)2488 3493 y Fz(,)28 b(whic)m(h)g(denotes)f Fy(')3226 3508 y Fx(F)3271 3517 y Fm(1)3310 3493 y Fz(\()p Fy(f)11 b Fz(\))456 3610 y(for)27 b Fy(f)38 b Fu(2)29 b Fy(F)858 3568 y Fw(\002)844 3634 y Fl(1)916 3610 y Fz(,)g(and)f(\()p Fy(f)11 b Fz(\))1292 3625 y Fx(G)1347 3636 y Fs(F)1386 3651 y Fm(1)1428 3610 y Fz(,)29= b(whic)m(h)h(denotes)f(the)g(pro)5 b(jection)28 b(of)g(res)17 b Fy(')3036 3625 y Fx(F)3095 3610 y Fz(\()p Fy(f)11 b Fz(\))27 b(on)m(to)456 3728 y(the)33 b(\014rst)g(summand.)555 3928= y(One)g(w)m(a)m(y)h(of)e(in)m (terpreting)i(this)f(restriction)h(map)f(is)g(with)g(the)g(follo)m (wing)456 4102 y FF(Lemma)c(7.)36 b Fp(L)-5 b(et)27 b Fy(G)1203 4117 y Fx(F)1290 4102 y Fz(=3D)g Fy(G)1470 4117 y Fx(F)1515 4126 y Fm(1)1559 4102 y Fy(?)5 b(G)1690 4117 y Fx(F)1735 4126 y Fm(2)1801 4102 y Fp(b)-5 b(e)27 b(pr)-5 b(o-)p Fy(p)27 b Fp(absolute)g(Galois)g(gr)-5 b(oups)27 b(of)g(\014elds)456 4218 y(c)-5 b(ontaining)35 b(a)g Fy(p)p Fp(th)i(r)-5 b(o)g(ot)36 b(of)g(unity,)g(and)g(supp)-5 b(ose)35 b(that)i(we)e(have)h(the)g(fol)5 b(lowing)456 4335 y(se)-5 b(quenc)g(e:)1416 4476 y Fy(G)1493 4491 y Fx(F)1616 4420 y Fl(can)1751 4449 y Fg(/)p Ff(/)29 b Fg(/)p Ff(/)p 1580 4451 200 4 v 1809 4471 a Fy(G)1886 4486 y Fx(F)1931 4495 y Fm(1)2168 4449 y Fg(/)p Ff(/)g Fg(/)p Ff(/)p 1998 4451 V 2226 4474 a Fv(Z)p Fy(=3Dp)p Fv(Z)p Fy(:)456 4626 y Fp(Her)-5 b(e)33 b(the)g(c)-5 b(anonic)g(al)32 b(map)h Fz(can)h Fp(is)f(an)g(identity)g(on)g Fy(G)2477 4641 y Fx(F)2522 4650 y Fm(1)2594 4626 y= Fp(and)g(c)-5 b(ontains)32 b Fy(G)3243 4641 y Fx(F)3288 4650 y Fm(2)3360 4626 y Fp(in)456 4743 y(its)j(kernel.)555 4942= y(Then)44 b(the)g(right-hand)g(surje)-5 b(ction)44= b(and)f(the)i(c)-5 b(omp)g(ose)g(d)43 b(surje)-5 b(ction)44 b(c)-5 b(orr)g(e-)456 5058 y(sp)g(ond)44 b(to)i(\014elds)e Fy(K)1210 5073 y Fl(1)1297 5058 y Fz(=3D)j Fy(F)1483 5073 y Fl(1)1523 5058 y Fz(\()1588 5034 y Fs(p)1565 4994 y Fu(p)p 1648 4994 91 4 v 64 x Fy(a)1699 5073 y Fl(1)1739 5058 y Fz(\))e Fp(and)g Fy(K)54 b Fz(=3D)47 b Fy(F)14 b Fz(\()2425 5025 y Fs(p)2401 4987 y Fu(p)p 2484 4987 52 4 v 71 x Fy(a)q Fz(\))p Fp(,)48 b(r)-5 b(esp)g(e)g(ctively,)47 b(wher)-5 b(e)456 5175 y Fz(\()p Fy(a)p Fz(\))583 5190 y Fx(G)638 5201 y Fs(F)677 5216 y Fm(1)747 5175 y Fz(=3D)27 b(\()p Fy(a)939 5190 y Fl(1)979 5175 y Fz(\))1017 5190 y Fx(F)1062 5199 y Fm(1)1135 5175 y Fp(and)34 b Fz(\()p Fy(a)p Fz(\))1451 5190 y Fx(G)1506 5201 y Fs(F)1545 5216 y Fm(2)1616 5175 y Fz(=3D)27 b(0)p Fp(.)p eop %%Page: 10 10 10 9 bop 456 255 a Fq(10)759 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456 450 y Fp(Pr)k(o)g(of.)41 b Fz(The)27= b(surjections)g(are)f(con)m(tin)m (uous)i(homomorphisms,)h(hence)e(elemen)m(ts)456 566 y(of)c Fy(H)647 530 y Fl(1)686 566 y Fz(\()p Fy(G)801 581 y Fx(F)846 590 y Fm(1)885 566 y Fy(;)17 b Fv(F)990 581 y Fx(p)1029 566 y Fz(\))24 b(and)g Fy(H)1361 530 y Fl(1)1400 566 y Fz(\()p Fy(G)1515 581 y Fx(F)1574 566 y Fy(;)17 b Fv(F)1679 581 y Fx(p)1718 566 y= Fz(\),)26 b(resp)s(ectiv)m(ely)-8 b(,)29 b(and)24 b(the)h(righ)m(t-hand)f (surjec-)456 683= y(tion)e(is)g(clearly)i(the)e(restriction)h(of)f(the)h (comp)s(osed)g(surjection.)41 b(The)23 b(remainder)456 799 y(follo)m(ws)33 b(b)m(y)g(Kummer)h(theory)-8 b(.)1757 b Fi(\003)456 980 y FF(Lemma)45 b(8.)f Fp(L)-5 b(et)40 b Fy(G)1240 995 y Fx(F)1335 980 y Fz(=3D)c Fy(G)1524 995 y Fx(F)1569 1004 y Fm(1)1633 980 y Fy(?)25 b(G)1784 995 y Fx(F)1829 1004 y Fm(2)1907 980 y Fp(b)-5 b(e)39 b(pr)-5 b(o-)p Fy(p)39 b Fp(absolute)h(Galois)f(gr)-5 b(oups)39 b(and)456 1097 y(supp)-5 b(ose)34 b(that)h Fz(\()p Fy(a)p Fz(\))1138 1112 y Fx(F)1232 1097 y Fp(and)f Fz(\()p Fy(b)p Fz(\))1538 1112 y Fx(F)1632 1097 y Fp(satisfy)h Fz(\()p Fy(a)p Fz(\))2063 1112 y Fx(G)2118 1123 y Fs(F)2157 1138 y Fm(1)2227 1097 y Fz(=3D)27 b(\()p Fy(b)p Fz(\))2447 1112 y Fx(G)2502 1123 y Fs(F)2541 1138 y Fm(2)2612 1097 y Fz(=3D)g(0)p Fp(.)45 b(Then)1308 1283 y Fz(\()p Fy(a)p Fz(\))1435 1298 y Fx(F)1515 1283 y Fu([)23 b Fz(\()p Fy(b)p Fz(\))1721 1298 y Fx(F)1808 1283 y Fz(=3D)k(0)h Fu(2)g Fy(H)2171 1242 y Fl(2)2210 1283 y Fz(\()p Fy(G)2325 1298 y Fx(F)2383 1283 y Fy(;)17 b Fv(F)2488 1298 y Fx(p)2527 1283 y Fz(\))p Fy(:)555 1553 y Fz(The)45= b(lemma)f(follo)m(ws)g(from)f ([N,)j(\(4.1\))d(Satz])h(and)f(from)g([Ris)q(,)j(Prop.)e(7.3,)456 1669 y(page)31 b(191].)43 b(Ho)m(w)m(ev)m(er,)34= b(w)m(e)f(pro)m(v)m(e) g(our)f(lemma)g(b)m(y)h(translating)f(the)g(cup)g(pro)s(d-)456 1786= y(ucts)d(in)m(to)f(obstructions)h(to)f(basic)h(Galois)f(em)m(b)s (edding)i(problems,)g(yielding)f(an)456 1902= y(in)m(teresting)34 b(Galois-theoretic)f(v)-5 b(arian)m(t)32 b(of)g(the)h(pro)s(of.)456 2172 y Fp(Pr)-5 b(o)g(of.)41 b Fz(If)d(\()p Fy(a)p= Fz(\))g(=3D)g(0)g(or)g (\()p Fy(b)p Fz(\))h(=3D)e(0,)j(w)m(e)g(are)f(done.)62 b(Otherwise,)41 b(the)e(conditions)456 2288 y(\()p Fy(a)p Fz(\))583 2303 y Fx(G)638 2314 y Fs(F)677 2329 y Fm(1)747 2288 y Fz(=3D)27 b(\()p Fy(b)p Fz(\))967 2303 y Fx(G)1022 2314 y Fs(F)1061 2329 y Fm(2)1132 2288 y Fz(=3D)g(0)32 b(imply)i(that)e(\()p Fy(a)p Fz(\))g(and)h(\()p= Fy(b)p Fz(\))f(are)g(linearly)i(indep)s(enden)m(t)g(in)456 2416 y Fy(H)545 2380 y Fl(1)584 2416 y Fz(\()p Fy(G)699 2431 y Fx(F)757 2416 y Fy(;)17 b Fv(F)862 2431 y Fx(p)901 2416 y Fz(\).)555 2621 y(No)m(w)33 b(let)g Fy(H)999 2637 y Fx(p)1035 2619 y Fm(3)1106 2621 y= Fz(b)s(e)g(the)g(Heisen)m(b)s(erg)h (group)e(of)h(order)f Fy(p)2594 2585 y Fl(3)2634 2621 y Fz(:)897 2790 y Fy(H)978 2807 y Fx(p)1014 2788 y Fm(3)1079 2790 y Fz(=3D)c Fu(h)p Fy(v)1269 2805 y Fl(1)1308 2790 y Fy(;)17 b(v)1399 2805 y Fl(2)1438 2790 y Fy(;)g(w)35 b Fu(j)d Fy(v)1698 2743 y Fx(p)1694 2815 y Fl(1)1765 2790 y Fz(=3D)c Fy(v)1920 2743 y Fx(p)1916 2815 y Fl(2)1987 2790 y Fz(=3D)g Fy(w)2164 2749 y Fx(p)2230 2790 y Fz(=3D)g(1)p Fy(;)17 b(v)2474 2805 y Fl(2)2513 2790 y Fy(v)2560 2805 y Fl(1)2627 2790 y Fz(=3D)28 b Fy(w)s(v)2851 2805 y Fl(1)2890 2790 y Fy(v)2937 2805 y Fl(2)2976 2790 y Fy(;)1222 2941 y Fz([)p Fy(v)1296 2956 y Fl(1)1335 2941 y Fy(;)17 b(w)s Fz(])27 b(=3D)h([)p Fy(v)1684 2956 y Fl(2)1723 2941 y Fy(;)17 b(w)s Fz(])27 b(=3D)h(1)p Fu(i)456 3111 y Fz(In)33 b(the)g(case)g Fy(p)28 b Fz(=3D)f(2,)33 b Fy(H)1322 3126 y Fl(8)1393 3111 y Fz(is)g(the)g(familiar)g(dihedral)g(group)g Fy(D)2757 3126 y Fl(4)2796 3111 y Fz(.)555 3316 y(By)41 b([M)q(,)h(Corollary)-8 b(,)43 b(page)d(523)g(and)h(Theorem)h(3\(A\)],)e(if)g(\()p Fy(a)p Fz(\))h(and)g(\()p Fy(b)p Fz(\))f(are)456 3432 y(linearly)31 b(indep)s(enden)m(t,)i(then)e(\()p Fy(a)p Fz(\))18 b Fu([)g Fz(\()p Fy(b)p Fz(\))28 b(=3D)f(0)j(if)g(and)h(only)g (if)f Fy(H)2812 3449 y Fx(p)2848 3430 y Fm(3)2916 3432 y Fz(is)h(the)g(Galois)456 3563 y(group)i(Gal)o(\()p Fy(M)5 b(=3DF)14 b Fz(\))33 b(of)g(a)g(Galois)g(extension)j Fy(M)44 b Fz(of)33 b Fy(F)47 b Fz(con)m(taining)34 b Fy(F)14 b Fz(\()3123 3529 y Fs(p)3100 3491 y Fu(p)p 3183 3491 52 4 v 72 x Fy(a;)3305 3514 y Fs(p)3282 3478 y Fu(p)p 3365 3478 42 4 v 85 x Fy(b)p Fz(\))456 3679 y(in)32= b(suc)m(h)i(a)f(w)m (a)m(y)h(that)570 3858 y Fy(H)651 3875 y Fx(p)687 3856 y Fm(3)725 3858 y Fy(=3D)p Fu(h)p Fy(v)860 3873 y Fl(1)899 3858 y Fy(;)17 b(w)s Fu(i)27 b Fz(=3D)g(Gal\()p Fy(F)14 b Fz(\()1518 3819 y Fs(p)1495 3781 y Fu(p)p 1578 3781 52 4 v 77 x Fy(a)p Fz(\))p Fy(=3DF)g Fz(\))32 b(and)g Fy(H)2133 3875 y Fx(p)2169 3856 y Fm(3)2208 3858 y Fy(=3D)p Fu(h)p Fy(v)2343 3873 y Fl(2)2382 3858 y Fy(;)17 b(w)s Fu(i)26 b Fz(=3D)i(Gal)o(\()p Fy(F)14 b Fz(\()3000 3803 y Fs(p)2977 3768 y Fu(p)p 3060 3768 42 4 v 90 x Fy(b)q Fz(\))p Fy(=3DF)g Fz(\))p Fy(:)555 4116 y Fz(No)m(w)33= b(consider)h(the)f(comm)m(utativ)m (e)i(diagram)1882 4294 y Fy(G)1959 4309 y Fx(F)1658 4402 y(\016)1689 4411 y Fm(1)1648 4519 y Fg(|)p Ff(|)1625 4537 y Fg(|)p Ff(|)1625 4538 y Fe(z)1655 4513 y(z)1685 4488 y(z)1715 4464 y(z)1744 4439 y(z)1774 4414 y(z)1804 4389 y(z)1833 4365 y(z)1979 4633 y Fx(\014)1950 4859 y Fg(\017)p Ff(\017)1950 4888 y Fg(\017)p Ff(\017)p 1948 4890 4 4 v 1948 4873 V 1948 4857 V 1948 4841 V 1948 4825 V 1948 4809 V 1948 4793 V 1948 4776 V 1948 4760 V 1948 4744 V 1948 4728 V 1948 4712 V 1948 4696 V 1948 4679 V 1948 4663 V 1948 4647 V 1948 4631 V 1948 4615 V 1948 4599 V 1948 4582 V 1948 4566 V 1948 4550 V 1948 4534 V 1948 4518 V 1948 4502 V 1948 4485 V 1948 4469 V 1948 4453 V 1948 4437 V 1948 4421 V 1948 4404 V 1948 4388 V 1948 4372 V 1948 4356 V 1948 4340 V 2176 4402 a Fx(\016)2207 4411 y Fm(2)2253 4519 y Fg(")p Ff(")2275 4537 y Fg(")p Ff(")2243 4511 y Fe(D)2213 4486 y(D)2184 4461 y(D)2154 4437 y(D)2124 4412 y(D)2094 4387 y(D)2065 4363 y(D)2035 4338 y(D)1455 4634 y Fy(G)1532 4649 y Fx(F)1577 4658 y Fm(1)1215 4751 y Fx(\013)1260 4760 y Fm(1)1206 4869 y Fg({)p Ff({)1183 4887 y Fg({)p Ff({)p Fe(x)1215 4862 y(x)1247 4837 y(x)1279 4812 y(x)1310 4788 y(x)1342 4763 y(x)1374 4738 y(x)1406 4713 y(x)2285 4634 y Fy(G)2362 4649 y Fx(F)2407 4658 y Fm(2)2605 4751 y Fx(\013)2650 4760 y Fm(2)2694 4869 y Fg(#)p Ff(#)2717 4887 y Fg(#)p Ff(#)2684 4861 y Fe(F)2652 4837 y(F)2621 4812 y(F)2589 4787 y(F)2557 4762 y(F)2525 4737 y(F)2493 4712 y(F)2461 4688 y(F)967 4991 y Fv(Z)p Fy(=3Dp)p Fv(Z)1206 4945 y Fd(\037)1226 4924 y(\177)1429 4927 y Fl(1)p Fw(7!)p Fx(v)1569 4936 y Fm(1)1843 4966 y Fg(/)p Ff(/)p 1226 4967 617 4 v 1872 4985 a Fy(H)1953 5002 y Fx(p)1989 4983 y Fm(3)2120 5203 y Fw(h)p Fx(v)2181 5212 y Fm(1)2216 5203 y Fx(;w)r Fw(i7!)p Fl(0;)c Fx(v)2507 5212 y Fm(2)2542 5203 y Fw(7!)p Fl(1)2712 5045 y Fg(8)p Ff(8)p 2053 5046 4 4 v 2055 5047 V 2056 5048 V 2058 5048 V 2060 5049 V 2061 5050 V 2063 5051 V 2064 5052 V 2066 5053 V 2068 5054 V 2069 5054 V 2071 5055 V 2072 5056 V 2074 5057 V 2076 5058 V 2077 5058 V 2079 5059 V 2081 5060 V 2082 5061 V 2084 5062 V 2085 5062 V 2087 5063 V 2089 5064 V 2090 5065 V 2092 5066 V 2094 5066 V 2095 5067 V 2097 5068 V 2099 5069 V 2100 5069 V 2102 5070 V 2104 5071 V 2106 5071 V 2107 5072 V 2109 5073 V 2111 5074 V 2112 5074 V 2114 5075 V 2116 5076 V 2117 5076 V 2119 5077 V 2121 5078 V 2123 5078 V 2124 5079 V 2126 5080 V 2128 5081 V 2129 5081 V 2131 5082 V 2133 5082 V 2135 5083 V 2136 5084 V 2138 5084 V 2140 5085 V 2142 5086 V 2143 5086 V 2145 5087 V 2147 5088 V 2149 5088 V 2150 5089 V 2152 5089 V 2154 5090 V 2156 5090 V 2157 5091 V 2159 5092 V 2161 5092 V 2163 5093 V 2165 5093 V 2166 5094 V 2168 5094 V 2170 5095 V 2172 5096 V 2174 5096 V 2175 5097 V 2177 5097 V 2179 5098 V 2181 5098 V 2183 5099 V 2184 5099 V 2186 5100 V 2188 5100 V 2190 5101 V 2192 5101 V 2193 5102 V 2195 5102 V 2197 5103 V 2199 5103 V 2201 5104 V 2203 5104 V 2204 5105 V 2206 5105 V 2208 5105 V 2210 5106 V 2212 5106 V 2214 5107 V 2215 5107 V 2217 5108 V 2219 5108 V 2221 5108 V 2223 5109 V 2225 5109 V 2226 5110 V 2228 5110 V 2230 5110 V 2232 5111 V 2234 5111 V 2236 5112 V 2238 5112 V 2240 5112 V 2241 5113 V 2243 5113 V 2245 5113 V 2247 5114 V 2249 5114 V 2251 5114 V 2253 5115 V 2254 5115 V 2256 5115 V 2258 5116 V 2260 5116 V 2262 5116 V 2264 5117 V 2266 5117 V 2268 5117 V 2270 5117 V 2271 5118 V 2273 5118 V 2275 5118 V 2277 5119 V 2279 5119 V 2281 5119 V 2283 5119 V 2285 5120 V 2287 5120 V 2288 5120 V 2290 5120 V 2292 5121 V 2294 5121 V 2296 5121 V 2298 5121 V 2300 5121 V 2302 5122 V 2304 5122 V 2306 5122 V 2307 5122 V 2309 5122 V 2311 5123 V 2313 5123 V 2315 5123 V 2317 5123 V 2319 5123 V 2321 5123 V 2323 5124 V 2325 5124 V 2327 5124 V 2329 5124 V 2330 5124 V 2332 5124 V 2334 5124 V 2336 5125 V 2338 5125 V 2340 5125 V 2342 5125 V 2344 5125 V 2346 5125 V 2348 5125 V 2350 5125 V 2352 5125 V 2353 5125 V 2355 5125 V 2357 5125 V 2359 5126 V 2361 5126 V 2363 5126 V 2365 5126 V 2367 5126 V 2369 5126 V 2371 5126 V 2373 5126 V 2375 5126 V 2377 5126 V 2378 5126 V 2380 5126 V 2382 5126 V 2384 5126 V 2386 5126 V 2388 5126 V 2390 5126 V 2392 5126 V 2394 5126 V 2396 5126 V 2398 5126 V 2400 5126 V 2402 5126 V 2403 5126 V 2405 5126 V 2407 5125 V 2409 5125 V 2411 5125 V 2413 5125 V 2415 5125 V 2417 5125 V 2419 5125 V 2421 5125 V 2423 5125 V 2425 5125 V 2426 5125 V 2428 5125 V 2430 5124 V 2432 5124 V 2434 5124 V 2436 5124 V 2438 5124 V 2440 5124 V 2442 5124 V 2444 5123 V 2446 5123 V 2448 5123 V 2449 5123 V 2451 5123 V 2453 5123 V 2455 5122 V 2457 5122 V 2459 5122 V 2461 5122 V 2463 5122 V 2465 5121 V 2467 5121 V 2469 5121 V 2470 5121 V 2472 5121 V 2474 5120 V 2476 5120 V 2478 5120 V 2480 5120 V 2482 5119 V 2484 5119 V 2486 5119 V 2488 5119 V 2489 5118 V 2491 5118 V 2493 5118 V 2495 5117 V 2497 5117 V 2499 5117 V 2501 5117 V 2503 5116 V 2504 5116 V 2506 5116 V 2508 5115 V 2510 5115 V 2512 5115 V 2514 5114 V 2516 5114 V 2518 5114 V 2519 5113 V 2521 5113 V 2523 5113 V 2525 5112 V 2527 5112 V 2529 5112 V 2531 5111 V 2533 5111 V 2534 5110 V 2536 5110 V 2538 5110 V 2540 5109 V 2542 5109 V 2544 5108 V 2546 5108 V 2547 5108 V 2549 5107 V 2551 5107 V 2553 5106 V 2555 5106 V 2557 5105 V 2558 5105 V 2560 5105 V 2562 5104 V 2564 5104 V 2566 5103 V 2568 5103 V 2569 5102 V 2571 5102 V 2573 5101 V 2575 5101 V 2577 5100 V 2578 5100 V 2580 5099 V 2582 5099 V 2584 5098 V 2586 5098 V 2587 5097 V 2589 5097 V 2591 5096 V 2593 5096 V 2595 5095 V 2596 5094 V 2598 5094 V 2600 5093 V 2602 5093 V 2604 5092 V 2605 5092 V 2607 5091 V 2609 5090 V 2611 5090 V 2612 5089 V 2614 5089 V 2616 5088 V 2618 5088 V 2619 5087 V 2621 5086 V 2623 5086 V 2625 5085 V 2626 5084 V 2628 5084 V 2630 5083 V 2632 5082 V 2633 5082 V 2635 5081 V 2637 5081 V 2639 5080 V 2640 5079 V 2642 5078 V 2644 5078 V 2645 5077 V 2647 5076 V 2649 5076 V 2651 5075 V 2652 5074 V 2654 5074 V 2656 5073 V 2657 5072 V 2659 5071 V 2661 5071 V 2662 5070 V 2664 5069 V 2666 5069 V 2667 5068 V 2669 5067 V 2671 5066 V 2672 5066 V 2674 5065 V 2676 5064 V 2677 5063 V 2679 5062 V 2681 5062 V 2682 5061 V 2684 5060 V 2686 5059 V 2687 5058 V 2689 5058 V 2691 5057 V 2692 5056 V 2694 5055 V 2695 5054 V 2697 5054 V 2699 5053 V 2700 5052 V 2702 5051 V 2703 5050 V 2705 5049 V 2707 5048 V 2708 5048 V 2710 5047 V 1252 5203 a Fw(h)p Fx(v)1313 5212 y Fm(2)1348 5203 y Fx(;w)r Fw(i7!)p Fl(0;)h Fx(v)1640 5212 y Fm(1)1675 5203 y Fw(7!)p Fl(1)1188 5045 y Fg(f)p Ff(f)p 1843 5046 V 1842 5047 V 1840 5048 V 1839 5048 V 1837 5049 V 1836 5050 V 1834 5051 V 1832 5052 V 1831 5053 V 1829 5054 V 1828 5054 V 1826 5055 V 1824 5056 V 1823 5057 V 1821 5058 V 1819 5058 V 1818 5059 V 1816 5060 V 1814 5061 V 1813 5062 V 1811 5062 V 1810 5063 V 1808 5064 V 1806 5065 V 1805 5066 V 1803 5066 V 1801 5067 V 1800 5068 V 1798 5069 V 1796 5069 V 1795 5070 V 1793 5071 V 1791 5071 V 1789 5072 V 1788 5073 V 1786 5074 V 1784 5074 V 1783 5075 V 1781 5076 V 1779 5076 V 1778 5077 V 1776 5078 V 1774 5078 V 1772 5079 V 1771 5080 V 1769 5081 V 1767 5081 V 1765 5082 V 1764 5082 V 1762 5083 V 1760 5084 V 1759 5084 V 1757 5085 V 1755 5086 V 1753 5086 V 1752 5087 V 1750 5088 V 1748 5088 V 1746 5089 V 1744 5089 V 1743 5090 V 1741 5090 V 1739 5091 V 1737 5092 V 1736 5092 V 1734 5093 V 1732 5093 V 1730 5094 V 1729 5094 V 1727 5095 V 1725 5096 V 1723 5096 V 1721 5097 V 1720 5097 V 1718 5098 V 1716 5098 V 1714 5099 V 1712 5099 V 1711 5100 V 1709 5100 V 1707 5101 V 1705 5101 V 1703 5102 V 1701 5102 V 1700 5103 V 1698 5103 V 1696 5104 V 1694 5104 V 1692 5105 V 1690 5105 V 1689 5105 V 1687 5106 V 1685 5106 V 1683 5107 V 1681 5107 V 1679 5108 V 1678 5108 V 1676 5108 V 1674 5109 V 1672 5109 V 1670 5110 V 1668 5110 V 1666 5110 V 1665 5111 V 1663 5111 V 1661 5112 V 1659 5112 V 1657 5112 V 1655 5113 V 1653 5113 V 1652 5113 V 1650 5114 V 1648 5114 V 1646 5114 V 1644 5115 V 1642 5115 V 1640 5115 V 1638 5116 V 1637 5116 V 1635 5116 V 1633 5117 V 1631 5117 V 1629 5117 V 1627 5117 V 1625 5118 V 1623 5118 V 1621 5118 V 1620 5119 V 1618 5119 V 1616 5119 V 1614 5119 V 1612 5120 V 1610 5120 V 1608 5120 V 1606 5120 V 1604 5121 V 1603 5121 V 1601 5121 V 1599 5121 V 1597 5121 V 1595 5122 V 1593 5122 V 1591 5122 V 1589 5122 V 1587 5122 V 1585 5123 V 1583 5123 V 1582 5123 V 1580 5123 V 1578 5123 V 1576 5123 V 1574 5124 V 1572 5124 V 1570 5124 V 1568 5124 V 1566 5124 V 1564 5124 V 1562 5124 V 1560 5125 V 1559 5125 V 1557 5125 V 1555 5125 V 1553 5125 V 1551 5125 V 1549 5125 V 1547 5125 V 1545 5125 V 1543 5125 V 1541 5125 V 1539 5125 V 1537 5126 V 1536 5126 V 1534 5126 V 1532 5126 V 1530 5126 V 1528 5126 V 1526 5126 V 1524 5126 V 1522 5126 V 1520 5126 V 1518 5126 V 1516 5126 V 1514 5126 V 1512 5126 V 1511 5126 V 1509 5126 V 1507 5126 V 1505 5126 V 1503 5126 V 1501 5126 V 1499 5126 V 1497 5126 V 1495 5126 V 1493 5126 V 1491 5126 V 1489 5125 V 1487 5125 V 1486 5125 V 1484 5125 V 1482 5125 V 1480 5125 V 1478 5125 V 1476 5125 V 1474 5125 V 1472 5125 V 1470 5125 V 1468 5125 V 1466 5124 V 1464 5124 V 1463 5124 V 1461 5124 V 1459 5124 V 1457 5124 V 1455 5124 V 1453 5123 V 1451 5123 V 1449 5123 V 1447 5123 V 1445 5123 V 1443 5123 V 1441 5122 V 1440 5122 V 1438 5122 V 1436 5122 V 1434 5122 V 1432 5121 V 1430 5121 V 1428 5121 V 1426 5121 V 1424 5121 V 1422 5120 V 1421 5120 V 1419 5120 V 1417 5120 V 1415 5119 V 1413 5119 V 1411 5119 V 1409 5119 V 1407 5118 V 1405 5118 V 1403 5118 V 1402 5117 V 1400 5117 V 1398 5117 V 1396 5117 V 1394 5116 V 1392 5116 V 1390 5116 V 1388 5115 V 1387 5115 V 1385 5115 V 1383 5114 V 1381 5114 V 1379 5114 V 1377 5113 V 1375 5113 V 1373 5113 V 1372 5112 V 1370 5112 V 1368 5112 V 1366 5111 V 1364 5111 V 1362 5110 V 1360 5110 V 1359 5110 V 1357 5109 V 1355 5109 V 1353 5108 V 1351 5108 V 1349 5108 V 1347 5107 V 1346 5107 V 1344 5106 V 1342 5106 V 1340 5105 V 1338 5105 V 1336 5105 V 1335 5104 V 1333 5104 V 1331 5103 V 1329 5103 V 1327 5102 V 1326 5102 V 1324 5101 V 1322 5101 V 1320 5100 V 1318 5100 V 1316 5099 V 1315 5099 V 1313 5098 V 1311 5098 V 1309 5097 V 1307 5097 V 1306 5096 V 1304 5096 V 1302 5095 V 1300 5094 V 1298 5094 V 1297 5093 V 1295 5093 V 1293 5092 V 1291 5092 V 1290 5091 V 1288 5090 V 1286 5090 V 1284 5089 V 1283 5089 V 1281 5088 V 1279 5088 V 1277 5087 V 1275 5086 V 1274 5086 V 1272 5085 V 1270 5084 V 1268 5084 V 1267 5083 V 1265 5082 V 1263 5082 V 1262 5081 V 1260 5081 V 1258 5080 V 1256 5079 V 1255 5078 V 1253 5078 V 1251 5077 V 1250 5076 V 1248 5076 V 1246 5075 V 1244 5074 V 1243 5074 V 1241 5073 V 1239 5072 V 1238 5071 V 1236 5071 V 1234 5070 V 1233 5069 V 1231 5069 V 1229 5068 V 1228 5067 V 1226 5066 V 1224 5066 V 1223 5065 V 1221 5064 V 1219 5063 V 1218 5062 V 1216 5062 V 1214 5061 V 1213 5060 V 1211 5059 V 1209 5058 V 1208 5058 V 1206 5057 V 1204 5056 V 1203 5055 V 1201 5054 V 1200 5054 V 1198 5053 V 1196 5052 V 1195 5051 V 1193 5050 V 1192 5049 V 1190 5048 V 1188 5048 V 1187 5047 V 2703 4991 a Fv(Z)p Fy(=3Dp)p Fv(Z)2674 4966 y Fd(?)2694 4945 y(_)2297 4927 y Fl(1)p Fw(7!)p Fx(v)2437 4936 y Fm(2)2057 4966 y Fg(o)p Ff(o)p 2057 4967 617 4 v eop %%Page: 11 11 11 10 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)292 b(11)456 450 y Fz(Let)35 b Fy(a)684 465 y Fl(2)755 450 y Fu(2)d Fy(F)930 409 y Fw(\002)916 475 y Fl(2)1024 450 y Fz(and)j Fy(b)1257 465 y Fl(1)1329 450 y Fu(2)d Fy(F)1504 409 y Fw(\002)1490 475 y Fl(1)1597 450 y Fz(satisfy)k(\()p Fy(a)1993 465 y Fl(2)2033 450 y Fz(\))2071 465 y Fx(F)2116 474 y Fm(2)2186 450 y Fz(=3D)31 b(\()p Fy(a)p Fz(\))2420 465 y Fx(G)2475 476 y Fs(F)2514 491 y Fm(2)2592 450 y Fz(and)k(\()p Fy(b)2863 465 y Fl(1)2903 450 y Fz(\))2941 465 y Fx(F)2986 474 y Fm(1)3056 450 y Fz(=3D)c(\()p Fy(b)p Fz(\))3280 465 y Fx(G)3335 476 y Fs(F)3374 491 y Fm(1)3417 450 y Fz(.)456 585 y(Then)f(set)g Fy(K)939 600 y Fl(1)1006 585 y Fz(=3D)d Fy(F)1172 600 y Fl(1)1212 585 y Fz(\()1277 545 y Fs(p)1254 507 y Fu(p)p 1337 507 81 4 v 78 x Fy(b)1378 600 y Fl(1)1418 585 y Fz(\))i(and)g Fy(K)1754 600 y Fl(2)1821 585 y Fz(=3D)f Fy(F)1988 600 y Fl(2)2027 585 y Fz(\()2093 560 y Fs(p)2069 521 y Fu(p)p 2153 521 91 4 v 2153 585 a Fy(a)2204 600 y Fl(2)2243 585 y Fz(\);)j(these)f(are)f Fv(Z)p Fy(=3Dp)p Fv(Z)p Fz(-extensions)456 701 y(of)e Fy(F)625 716 y Fl(1)692 701 y Fz(and)h Fy(F)940 716 y Fl(2)979 701 y Fz(,)h(resp)s(ectiv)m(ely) -8 b(.)45 b(W)-8 b(e)28 b(ma)m(y)h(then)f(iden)m(tify)h(the)f (left-hand)g Fv(Z)p Fy(=3Dp)p Fv(Z)g Fz(in)456 817= y(the)f(diagram)f (with)h(Gal)o(\()p Fy(K)1480 832 y Fl(1)1520 817 y Fy(=3DF)1632 832 y Fl(1)1671 817 y Fz(\))g(so)f(that)h Fy(\013)2117 832 y Fl(1)2183 817 y Fz(is)g(the)g(surjection)g(of)f(Galois)g(the-)456 934 y(ory)-8 b(.)41 b(Similarly)-8 b(,)29 b(the)e(righ)m(t-hand)f Fv(Z)p Fy(=3Dp)p Fv(Z)h= Fz(ma)m(y)g(b)s(e)g(iden)m(ti\014ed)h(with)f (Gal\()p Fy(K)3216 949 y Fl(2)3255 934 y Fy(=3DF)3367 949 y Fl(2)3406 934 y Fz(\))456 1050 y(with)d Fy(\013)731 1065 y Fl(2)794 1050 y Fz(the)g(surjection)h(of)e(Galois)g(theory)-8 b(.)41 b(Finally)-8 b(,)26 b(the)e(topmost)g(surjections)456 1166 y Fy(\016)499 1181 y Fl(1)571 1166 y Fz(and)32 b Fy(\016)803 1181 y Fl(2)876 1166 y Fz(are)g(canonical.)555 1397 y(By)50 b(Lemma)g(7,)j(the)c(surjections)i Fy(\016)1949 1412 y Fl(1)1988 1397 y Fy(\013)2050 1412 y Fl(1)2139 1397 y Fz(and)e Fy(\016)2388 1412 y Fl(2)2428 1397 y Fy(\013)2490 1412 y Fl(2)2578 1397 y Fz(corresp)s(ond)h(to)f(\014elds) 456 1518 y Fy(F)14 b Fz(\()598 1468 y Fs(p)575 1433 y Fu(p)p 658 1433 42 4 v 85 x Fy(b)p Fz(\))35 b(and)g Fy(F)14 b Fz(\()1106 1484 y Fs(p)1083 1446 y Fu(p)p 1166 1446 52 4 v 72 x Fy(a)p Fz(\),)35 b(resp)s(ectiv)m(ely)-8 b(.)54 b(No)m(w)35 b(b)s(ecause)i Fy(G)2551 1533 y Fx(F)2641 1518 y Fz(=3D)31 b Fy(G)2825 1533 y Fx(F)2870 1542 y Fm(1)2932 1518 y Fy(?)24 b(G)3082 1533 y Fx(F)3127 1542 y Fm(2)3165 1518 y Fz(,)36 b(there)456 1634 y(exists)30 b(a)e(homomorphism)i Fy(\014)j Fz(:)28 b Fy(G)1703 1649 y Fx(F)1789 1634 y Fu(!)g Fy(H)1998 1651 y Fx(p)2034 1632 y Fm(3)2072 1634 y Fz(,)h(and)g(b)s(ecause)h Fy(v)2718 1649 y Fl(1)2786 1634 y Fz(and)f Fy(v)3019 1649 y Fl(2)3087 1634 y Fz(generate)456 1750 y Fy(H)537 1767 y Fx(p)573 1748 y Fm(3)611 1750 y Fz(,)j Fy(\014)38 b Fz(is)c(a)e(surjection.)555 1981 y(Hence)i Fy(H)926 1998 y Fx(p)962 1979 y Fm(3)1033 1981 y Fz(is)f(a)g(Galois)f(group)h(o)m(v)m(er)g Fy(F)47 b Fz(corresp)s(onding)33 b(to)g(a)f(normal)h(sub-)456 2098 y(group)45 b Fy(H)53 b Fz(of)46 b Fy(G)1081 2113 y Fx(F)1139 2098 y Fz(.)83 b(Consider)47= b(the)f(smallest)i(normal)d(subgroup)i Fy(H)3100 2113 y Fl(1)3185 2098 y Fz(of)e Fy(G)3386 2113 y Fx(F)456 2214 y Fz(con)m(taining)35 b Fy(H)41 b Fz(and)35 b Fy(G)1323 2229 y Fx(F)1368 2238 y Fm(2)1406 2214 y Fz(.)49 b(Then)35 b(b)m(y)g(the)g(diagram,)g Fy(G)2531 2229 y Fx(F)2589 2214 y Fy(=3DH)2719 2229 y Fl(1)2793 2214 y Fz(is)g(the)f(left-hand)456 2340 y Fv(Z)p Fy(=3Dp)p Fv(Z)p Fz(,)26 b(whic)m(h)g(corresp)s(onds)g(to)e Fy(F)14 b Fz(\()1790 2291 y Fs(p)1767 2255 y Fu(p)p 1850 2255 42 4 v 85 x Fy(b)p Fz(\),)26 b(and)f Fy(H)2245 2355 y Fl(1)2284 2340 y Fy(=3DH)31 b Fz(is)25 b Fu(h)p Fy(v)2621 2355 y Fl(2)2661 2340 y Fy(;)17 b(w)s Fu(i)p Fz(.)39 b(No)m(w)25 b(consider)456 2456 y(the)34 b(smallest)i(normal)e(subgroup)h Fy(H)1840 2471 y Fl(2)1914 2456 y Fz(of)f Fy(G)2104 2471 y Fx(F)2196 2456 y Fz(con)m(taining)h Fy(H)42 b Fz(and)34 b Fy(G)3063 2471 y Fx(F)3108 2480 y Fm(1)3147 2456 y Fz(.)48 b(Then)456 2572 y(b)m(y)34 b(the)f(diagram,)g Fy(G)1244 2587 y Fx(F)1303 2572 y Fy(=3DH)1433 2587 y Fl(2)1505 2572 y Fz(is)g(the)h(righ)m(t-hand)f Fv(Z)p Fy(=3Dp)p= Fv(Z)p Fz(,)h(whic)m(h)g(corresp)s(onds)h(to)456 2689 y Fy(F)14 b Fz(\()598 2655 y Fs(p)575 2617 y Fu(p)p 658 2617 52 4 v 72 x Fy(a)p Fz(\),)32 b(and)h Fy(H)1077 2704 y Fl(2)1116 2689 y Fy(=3DH)40 b Fz(is)33 b Fu(h)p Fy(v)1470 2704 y Fl(1)1510 2689 y Fy(;)17 b(w)s Fu(i)p Fz(.)555 2920 y(Hence)34 b(\()p Fy(a)p Fz(\))22 b Fu([)h Fz(\()p Fy(b)p Fz(\))28 b(=3D)f(0.)1960 b Fi(\003)555 3296 y Fz(Finally)-8= b(,)36 b(w)m(e)h(close)f(with)f(with)h(a)f(companion)h(to)f(Lemma)g(7.)51 b(In)36 b(Lemma)g(9)456 3413 y(b)s(elo)m(w,)c= Fy(\031)k Fz(denotes)c(the)g(canonical)g(homomorphism)h(of)e Fy(G)g Fz(to)h Fy(G)2896 3428 y Fl(1)2966 3413 y Fz(whic)m(h)h(is)f(an)456 3529 y(iden)m(tit)m(y)i(on)e Fy(G)1028 3544 y Fl(1)1068 3529 y Fz(,)g(and)h(is)g(trivial)g(on)g Fy(G)1918 3544 y Fl(2)1957 3529 y Fz(.)456 3748 y FF(Lemma)48 b(9.)e Fp(L)-5 b(et)43 b Fy(G)f Fz(=3D)g Fy(G)1485 3763 y Fl(1)1552 3748 y Fy(?)28 b(G)1706 3763 y Fl(2)1788 3748 y Fp(b)-5 b(e)42 b(a)g(fr)-5 b(e)g(e)42 b(pr)-5 b(o)g(duct)43 b(of)f Fy(G)2752 3763 y Fl(1)2834 3748 y Fp(and)g Fy(G)3108 3763 y Fl(2)3190 3748 y Fp(in)g(the)456 3864 y(c)-5= b(ate)g(gory)36 b(of)h(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)50 b(Supp)-5 b(ose)36 b(that)h Fy(A)2241 3836 y Fu(\030)2242 3868 y Fz(=3D)2350 3864 y Fv(Z)p Fy(=3Dp)p Fv(Z)g= Fp(is)g(a)f(factor)h(gr)-5 b(oup)37 b(of)456 3980 y Fy(G)533 3995 y Fl(1)602 3980 y Fp(such)29 b(that)i(the)f(surje)-5 b(ction)29 b Fy(G)1691 3995 y Fl(1)1758 3980 y Fu(!)e Fv(Z)p Fy(=3Dp)p Fv(Z)k Fp(do)-5 b(es)29 b(not)h(factor)g(thr)-5 b(ough)30 b Fv(Z)p Fy(=3Dp)3309 3944 y Fl(2)3348 3980 y Fv(Z)p Fp(.)456 4096 y(Then)k(the)h(fol)5 b(lowing)33 b(c)-5 b(ommutative)34 b(diagr)-5 b(am)34 b(c)-5 b(annot)35 b(o)-5 b(c)g(cur:)1725 4364 y Fy(G)1957 4301 y Fx(\031)2097 4330 y Fg(/)p Ff(/)29 b Fg(/)p Ff(/)p 1831 4331 296 4 v 1763 4570 a Fg(\017)p Ff(\017)1763 4599 y Fg(\017)p Ff(\017)p 1761 4599 4 207 v 2155 4357 a Fy(G)2232 4372 y Fl(1)2214 4588 y Fg(\017)p Ff(\017)2214 4617 y Fg(\017)p Ff(\017)p 2212 4617 4 218 v 1628 4707 a Fv(Z)p Fy(=3Dp)1792 4671 y Fl(2)1832 4707 y Fv(Z)2105 4680 y Fg(/)p Ff(/)h Fg(/)p Ff(/)p 1927 4682 208 4 v 2164 4715 a Fy(A:)456 5099 y Fp(Pr)-5 b(o)g(of.)41 b Fz(Supp)s(ose)33= b(that)f(con)m(trary)h(to)f(our)g(statemen)m(t,)i (suc)m(h)g(a)e(diagram)g(as)h(the)456 5216 y(ab)s(o)m(v)m(e)h(exists.) 50 b(Then)35= b(b)m(y)g(passing)g(to)e(quotien)m(ts)j(b)m(y)f(comm)m (utator)f(subgroups)p eop %%Page: 12 12 12 11 bop 456 255 a Fq(12)759 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456 450 y Fz(w)m(e)33 b(obtain)1673 626 y Fy(G)1750 590 y Fx(ab)1926 556 y(\031)1969 533 y Fs(ab)2081 585 y Fg(/)p Ff(/)c Fg(/)p Ff(/)p 1850 587 260 4 v 1673 781 a Fx(\013)1747 838 y Fg(\017)p Ff(\017)1747 867 y Fg(\017)p Ff(\017)p 1745 867 4 212 v 2139 614 a Fy(G)2216 578 y Fx(ab)2216 639 y Fl(1)2243 775 y Fx(\015)2214 856 y Fg(\017)p Ff(\017)2214 885 y Fg(\017)p Ff(\017)p 2212 885 4 218 v 1612 975 a Fv(Z)p Fy(=3Dp)1776 939 y Fl(2)1816 975 y Fv(Z)1959 906 y Fx(\014)2105 948 y Fg(/)p Ff(/)h Fg(/)p Ff(/)p 1911 950 224 4 v 2164 982 a Fy(A:)456 1151 y Fz(But)g Fy(G)724 1115 y Fx(ab)824 1123 y Fu(\030)825 1155 y Fz(=3D)929 1151 y Fy(G)1006 1115 y Fx(ab)1006 1175 y Fl(1)1096 1151 y Fu(\002)18 b Fy(G)1268 1115 y Fx(ab)1268 1175 y Fl(2)1370 1151 y Fz(and)31 b(the)g(canonical)g(surjection)h(on)m(to)e Fy(G)2890 1115 y Fx(ab)2890 1175 y Fl(1)2993 1151 y= Fz(is)h(giv)m(en)g (b)m(y)456 1267 y(the)38 b(pro)5 b(jection)39 b(map.)61 b(Let)38 b Fy(\016)k= Fz(b)s(e)d(a)e(splitting)i(map)g(of)e(the)i(pro)5 b(jection)39 b(map.)456 1383 y(Then)33 b Fy(\015)g Fz(=3D)28 b Fy(\014)6 b(\013)q(\016)t Fz(,)31= b(con)m(tradicting)j(the)f(h)m(yp)s (othesis.)1005 b Fi(\003)1305 1699 y Fz(3.)49 b= Fj(Pr)n(oof)38 b(of)g(the)g(Theorem)555 1961 y Fz(First)46 b(w)m(e)h(de\014ne)f (\014elds)h Fy(F)1583 1976 y Fl(0)1623 1961 y Fz(,)i Fy(F)1762 1976 y Fl(1)1801 1961 y Fz(,)g Fy(F)1940 1976 y Fl(2)1980 1961 y Fz(,)g(and)c Fy(F)60 b= Fz(using)46 b(our)f(giv)m(en)i(cardinal)456 2077 y(n)m(um)m(b)s(ers)26 b Fy(d)p Fz(,)f Fy(e)p= Fz(,)h(and)e(\007,)i(as)e(w)m(ell)i(as)e(the)g (prime)h(n)m(um)m(b)s(er)h Fy(p)p Fz(.)40 b(Then)25= b(w)m(e)h(de\014ne) f(the)456 2193 y(cyclic)32 b(Galois)f(extension)h Fy(K)r(=3DF)44 b Fz(of)30 b(degree)i Fy(p)f= Fz(and)g(c)m(hec)m(k)i(that)d(the)h (arithmetic)456 2309 y(in)m(v)-5 b(arian)m(ts)33 b(of)f Fy(K)r(=3DF)46 b Fz(coincide)34 b(with)f Fy(d)p Fz(,)g Fy(e)p Fz(,)f(and)h(\007.)456 2593 y(3.1.)48 b FF(Constructing)37 b Fy(F)1394 2608 y Fl(0)1434 2593 y FF(,)g Fy(F)1565 2608 y Fl(1)1605 2593 y FF(,)h Fy(F)1737 2608 y Fl(2)1776 2593 y FF(,)g(and)g Fy(F)14 b FF(.)555 2796 y= Fz(If)39 b(\007)f(=3D)h(1)f(then)i(let)f Fy(F)1414 2811 y Fl(0)1492 2796 y Fz(=3D)f Fv(C)p Fz(.)63 b(If)38 b(\007)h(=3D)f(0)h(and)g Fy(p)f Fz(=3D)g(2,)j(let)e Fy(F)2912 2811 y Fl(0)2990 2796 y Fz(=3D)f Fv(R)p Fz(.)63 b(\(See)456 2912= y(Prop)s(osition)33 b(1)f(for)h(alternativ)m(e)h(c)m(hoices)h(in)e(these)h(t)m(w)m(o)g (cases.\))45 b(If)33 b(\007)28 b(=3D)h(0)j(and)456 3029 y Fy(p)27 b(>)h Fz(2)i(then)h(let)g Fy(F)1137 3044 y Fl(0)p Fx(;p)1262 3029 y= Fz(b)s(e)g(the)g(algebraic)g(extension)h(of)e Fv(Q)h Fz(of)f(Lemma)h(5.)43 b(In)31 b(the)456 3145= y(\014rst)d(t)m(w)m (o)h(cases)g(w)m(e)g(see)g(trivially)g(that)f Fy(\030)1978 3160 y Fx(p)2045 3145 y Fu(2)g Fy(F)2202 3160 y Fl(0)2242 3145 y Fz(,)h(and)f(in)g(the)h(last)f(case)h Fy(\030)3181 3160 y Fx(p)3248 3145 y Fu(2)f Fy(F)3405 3160 y Fl(0)456 3261 y Fz(b)m(y)33 b(Lemma)g(6.)43 b(Observ)m(e)35 b(that)1077 3574 y(dim)1240 3589 y Ft(F)1281 3597 y Fs(p)1337 3574 y Fy(F)1414 3533 y Fw(\002)1400 3599 y Fl(0)1473 3574 y Fy(=3DF)1599 3527 y Fw(\002)p Fx(p)1585 3599 y Fl(0)1721 3574 y Fz(=3D)1824 3340 y Fo(8)1824 3430 y(>)1824 3460 y(<)1824 3639 y(>)1824 3669 y(:)1913 3438 y Fz(0)p Fy(;)49 b Fz(if)32 b(\007)c(=3D)f(1;)1913 3577 y(1)p Fy(;)49 b Fz(if)32 b(\007)c(=3D)f(0)p Fy(;)49 b(p)28 b Fz(=3D)f(2;)1913 3717 y(2)p Fy(;)49 b Fz(if)32 b(\007)c(=3D)f(0)p Fy(;)49 b(p)28 b(>)f Fz(2)p Fy(:)3320 3574 y Fz(\(2\))555 3975 y(W)-8 b(e)39 b(next)h(construct)g(the)f(\014eld)h Fy(F)1845 3990 y Fl(1)1885 3975 y Fz(.)62 b(Because)40 b(\007)f(=3D)f(0)g(implies)j (1)d Fu(\024)h Fy(d)p Fz(,)h(for)456 4091= y(either)35 b(c)m(hoice)g(of)f(\007)c Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)34 b Fz(there)g(exists)j(a)c(w)m(ell-ordered)j(set)f Fy(I)2970 4106 y Fl(1)3044 4091 y Fz(suc)m(h)h(that)456 4225 y Fu(j)p Fy(I)527 4240 y Fl(1)566 4225 y Fu(j)20 b Fz(+)h(1)28 b(=3D)f Fy(d)21 b Fz(+)g(2\007.)43 b(Let)32 b(\000)1490 4240 y Fl(1)1557 4225 y Fz(=3D)c Fv(Z)1727 4174 y Fl(\()p Fx(I)1785 4183 y Fm(1)1820 4174 y Fl(\))1727 4257 y(\()p Fx(p)p Fl(\))1883 4225 y Fz(b)s(e)33 b(a)e(direct)i(sum)g (of)f Fu(j)p Fy(I)2760 4240 y Fl(1)2799 4225 y Fu(j)g Fz(copies)h(of)e Fv(Z)3322 4241 y Fl(\()p Fx(p)p Fl(\))3417 4225 y Fz(.)456 4356 y(Then)c(\000)765 4371 y Fl(1)830 4356 y Fz(is)g(a)f(linearly)h(ordered)g(ab)s(elian)f(group.)41 b(Finally)27 b(set)g Fy(F)2865 4371 y Fl(1)2932 4356 y Fz(:=3D)h Fy(F)3126 4371 y Fl(0)3165 4356 y Fz(\(\(\000)3302 4371 y Fl(1)3341 4356 y Fz(\)\).)456 4472 y(F)-8= b(rom)32 b(Lemmas)h(2)g(and)f(3)h(it)f(follo)m(ws)h(that)g Fy(G)2157 4487 y Fx(F)2202 4496 y Fm(1)2273 4472 y Fz(is)g(a)f(pro-)p Fy(p)p Fz(-group)f(and)1126 4644 y(dim)1289 4659 y Ft(F)1330 4667 y Fs(p)1386 4644 y Fy(F)1463 4603 y Fw(\002)1449 4668 y Fl(1)1522 4644 y Fy(=3DF)1648 4596 y Fw(\002)p Fx(p)1634 4668 y Fl(1)1770 4644 y Fz(=3D)c(dim)2036 4659 y Ft(F)2077 4667 y Fs(p)2133 4644 y Fy(F)2210 4603 y Fw(\002)2196 4668 y Fl(0)2269 4644 y Fy(=3DF)2395 4596 y Fw(\002)p Fx(p)2381 4668 y Fl(0)2511 4644 y Fz(+)22 b Fu(j)p Fy(I)2680 4659 y Fl(1)2719 4644 y Fu(j)p Fy(:)456 4811 y Fz(Hence)1016 5064 y(dim)1179 5079 y Ft(F)1220 5087 y Fs(p)1276 5064 y Fy(F)1353 5023 y Fw(\002)1339 5089 y Fl(1)1412 5064 y Fy(=3DF)1538 5017 y Fw(\002)p Fx(p)1524 5089 y Fl(1)1660 5064 y Fz(=3D)1763 4830 y Fo(8)1763 4920 y(>)1763 4950 y(<)1763 5129 y(>)1763 5159 y(:)1852 4928 y Fy(d)g Fz(+)g(1)p Fy(;)97 b Fz(if)33 b(\007)27 b(=3D)h(1;)1852 5067 y Fy(d;)266 b Fz(if)33 b(\007)27 b(=3D)h(0)p Fy(;)49 b(p)27 b Fz(=3D)h(2;)1852 5207 y Fy(d)22 b Fz(+)g(1)p Fy(;)97 b Fz(if)33 b(\007)27 b(=3D)h(0)p Fy(;)49 b(p)27 b(>)h Fz(2)p Fy(:)3320 5064 y Fz(\(3\))p eop %%Page: 13 13 13 12 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)292 b(13)555 450 y Fz(Similarly)-8 b(,)35 b(w)m(e)g(construct)g Fy(F)1628 465 y Fl(2)1701 450 y Fz(as)f(follo)m(ws.)47 b(Because)36 b Fy(p)29= b(>)h Fz(2)j(and)h(also)g Fy(p)29 b Fz(=3D)h(2)456 566= y(and)37 b(\007)e(=3D)g(1)h(implies)j Fy(e)c(>)g= Fz(0,)j(there)f(exists)i(a)e(w)m (ell-ordered)h(set)g Fy(I)2965 581 y Fl(2)3041 566 y Fz(suc)m(h)h(that)456 683 y(1)23 b(+)h Fu(j)p Fy(I)699 698 y Fl(2)738 683 y Fu(j)31 b Fz(=3D)h Fy(e)j= Fz(in)g(either)h(of)e(the) i(cases)g Fy(p)31 b(>)h Fz(2)j(or)f(\007)e(=3D)f(1,)36 b Fy(p)31 b Fz(=3D)h(2,)j(and)g Fu(j)p Fy(I)3193 698 y Fl(2)3232 683 y Fu(j)d Fz(=3D)f Fy(e)456 817 y Fz(in)k(the)h(case)h(\007) 32 b(=3D)g(0,)k Fy(p)d Fz(=3D)f(2.)52 b(Then)37 b(again)e(\000)2181 832 y Fl(2)2253 817 y Fz(=3D)d Fv(Z)2427 766 y Fl(\()p Fx(I)2485 775 y Fm(2)2520 766 y Fl(\))2427 848 y(\()p Fx(p)p Fl(\))2587 817 y Fz(is)k(a)f(linearly)i(ordered)456 948 y(ab)s(elian)32 b(group.)44 b(W)-8 b(e)33 b(set)h Fy(F)1493 963 y Fl(2)1560 948 y Fz(:=3D)28 b Fv(C)p Fz(\(\(\000)1900 963 y Fl(2)1939 948 y Fz(\)\).)44 b(Then)33= b(from)g([K,)g(pages)g(3)f (and)h(4])f(it)456 1064 y(follo)m(ws)g(that)g Fy(G)1063 1079 y Fx(F)1108 1088 y Fm(2)1174 1036 y Fu(\030)1175 1068 y Fz(=3D)1279 1064 y Fv(Z)1345 1028 y Fx(I)1376 1037 y Fm(2)1345 1088 y Fx(p)1415 1064 y= Fz(,)g(the)g(top)s(ological)f(pro)s (duct)h(of)g Fu(j)p Fy(I)2683 1079 y Fl(2)2722 1064 y Fu(j)f Fz(copies)i(of)f Fv(Z)3245 1079 y Fx(p)3285 1064 y Fz(.)43 b(In)456 1180 y(particular)760 1469 y Fy(e)28 b Fz(=3D)936 1298 y Fo(\()1017 1402 y Fz(dim)1179 1417 y Ft(F)1220 1425 y Fs(p)1277 1402 y Fy(F)1354 1361 y Fw(\002)1340 1426 y Fl(2)1412 1402 y Fy(=3DF)1538 1355 y Fw(\002)p Fx(p)1524 1426 y Fl(2)1655 1402 y Fz(+)22 b(1)p Fy(;)97 b Fz(if)32 b Fy(p)c(>)g Fz(2)k(or)g Fy(p)c Fz(=3D)f(2)33 b(and)f(\007)c(=3D)f(1;)1017 1542 y(dim)1179 1557 y Ft(F)1220 1565 y Fs(p)1277 1542 y Fy(F)1354 1500 y Fw(\002)1340 1566 y Fl(2)1412 1542 y Fy(=3DF)1538 1494 y Fw(\002)p Fx(p)1524 1566 y Fl(2)1632 1542 y Fy(;)267 b Fz(if)32 b Fy(p)c Fz(=3D)g(2)p Fy(;)49 b Fz(\007)27 b(=3D)h(0)p Fy(:)3320 1469 y Fz(\(4\))555 1859 y(F)-8 b(rom)35= b(Theorem)h(2)f(w)m (e)h(see)g(that)f(there)g(exists)i(a)e(\014eld)g Fy(F)49 b Fz(of)35 b(c)m(haracteristic)456 1976 y(zero,)30= b(suc)m(h)h(that)e Fy(G)1188 1991 y Fx(F)1274 1976 y Fz(=3D)e Fy(G)1454 1991 y Fx(F)1499 2000 y Fm(1)1553 1976 y Fy(?)15 b(G)1694 1991 y Fx(F)1739 2000 y Fm(2)1807 1976 y Fz(is)29= b(a)g(free)h(pro)s (duct)f(of)g Fy(G)2710 1991 y Fx(F)2755 2000 y Fm(1)2823 1976 y Fz(and)g Fy(G)3086 1991 y Fx(F)3131 2000 y Fm(2)3198 1976 y Fz(in)h(the)456 2092 y(category)h(of)f(pro-)p Fy(p)p Fz(-groups.)42 b(In)31 b(particular)g Fy(G)2208 2107 y Fx(F)2297 2092 y Fz(is)g(again)f(a)g(pro-)p= Fy(p)p Fz(-group,)g(and)456 2208 y(from)i(Lemma)h(6)f(w)m(e)i(see)g(that)e Fy(F)46 b Fz(con)m(tains)34 b(a)e(primitiv)m(e)i= Fy(p)p Fz(th)f(ro)s(ot)f(of)g(unit)m(y)-8 b(.)456 2576 y(3.2.)48 b FF(Constructing)37 b Fy(K)r(=3DF)14 b FF(.)555 2794 y Fz(W)-8 b(e)26 b(de\014ne)g(the)g(cyclic)h(extension)g Fy(K)r(=3DF)38 b Fz(of)25 b(degree)h Fy(p)f Fz(as)g Fy(K)35 b Fz(=3D)28 b Fy(F)14 b Fz(\()3021 2761 y Fs(p)2998 2723 y Fu(p)p 3081 2723 52 4 v 71 x Fy(a)p Fz(\))25 b(where)456 2911 y([)p Fy(a)p Fz(])j Fu(2)g Fy(F)760 2874 y Fw(\002)819 2911 y Fy(=3DF)945 2874 y Fw(\002)p Fx(p)1071 2911 y= Fz(is)33 b(c)m(hosen)h(via)f(the)g(isomorphism)h(\(1\).)555 3129 y(If)39 b(\007)g(=3D)f(1)h(then)h(let)f(\()p Fy(a)1441 3144 y Fl(1)1481 3129 y Fz(\))1519 3144 y Fx(F)1564 3153 y Fm(1)1641 3129 y Fz(b)s(e)g(an)m(y)h(non)m(trivial)g(elemen)m(t)h(in) e Fy(H)2998 3093 y Fl(1)3037 3129 y Fz(\()p Fy(G)3152 3144 y Fx(F)3197 3153 y Fm(1)3235 3129 y Fy(;)17 b Fv(F)3340 3144 y Fx(p)3379 3129 y Fz(\),)456 3245= y(whic)m(h)35 b(is)g(p)s(ossible)g(since)h(1)30 b Fu(\024)h Fz(dim)1794 3260 y Ft(F)1835 3268 y Fs(p)1892 3245 y Fy(H)1981 3209 y Fl(1)2020 3245 y Fz(\()p Fy(G)2135 3260 y Fx(F)2180 3269 y Fm(1)2218 3245 y Fy(;)17 b Fv(F)2323 3260 y Fx(p)2362 3245 y Fz(\))30 b(=3D)g Fy(d)23 b Fz(+)g(1.)48 b(Let)34 b(\()p Fy(a)3098 3260 y Fl(2)3138 3245 y Fz(\))3176 3260 y Fx(F)3221 3269 y Fm(2)3290 3245 y Fz(=3D)c(0)456 3365 y(in)i Fy(H)658 3329 y Fl(2)697 3365 y Fz(\()p Fy(G)812 3380 y Fx(F)857 3389 y Fm(2)896 3365 y Fy(;)17 b Fv(F)1001 3380 y Fx(p)1040 3365 y Fz(\).)555 3583 y(No)m(w)42= b(supp)s(ose)h (\007)g(=3D)g(0)e(and)h Fy(p)h(>)g Fz(2.)70 b(W)-8= b(e)42 b(denote)h(the)f(\014xed)g(\014eld)h(of)e(the)456 3699 y(factor)34 b Fv(Z)803 3714 y Fx(p)877 3699 y Fz(in)h Fy(G)1070 3663 y Fx(ab)1070 3725 y(F)1115 3734 y Fm(0)1185 3672 y Fu(\030)1185 3704 y Fz(=3D)1293 3699 y Fv(Z)1359 3714 y Fx(p)1423 3699 y Fu(\002)24 b Fv(Z)p Fy(=3Dp)p Fv(Z)35 b Fz(acting)g(on)f(a)g(maximal)i(ab)s(elian)f(extension)456 3835 y Fy(F)533 3799 y Fx(ab)519 3860 y Fl(0)641 3835 y Fz(of)i Fy(F)820 3850 y Fl(0)897 3835 y Fz(as)h Fy(K)1105 3850 y Fl(0)1180 3835 y Fz(:=3D)e Fy(F)1382 3850 y Fl(0)1421 3835 y Fz(\()1487 3786 y Fs(p)1463 3750 y Fu(p)p 1546 3750 42 4 v 85 x Fy(b)q Fz(\).)58 b(\(See)38= b(the)f(discussion)j (follo)m(wing)e(Lemma)g(5.\))456 3956 y(Because)45 b Fy(F)917 3914 y Fw(\002)903 3980 y Fl(0)975 3956 y Fy(=3DF)1101 3908 y Fw(\002)p Fx(p)1087 3980 y Fl(0)1239 3956 y= Fz(is)f(naturally)g (isomorphic)g(to)f(a)g(subgroup)i(of)e Fy(F)3139 3914 y Fw(\002)3125 3980 y Fl(1)3197 3956 y Fy(=3DF)3323 3908 y Fw(\002)p Fx(p)3309 3980 y Fl(1)3417 3956 y Fz(,)456 4072 y(w)m(e)g(ma)m(y)g(set)h(\()p Fy(a)1082 4087 y Fl(1)1121 4072 y Fz(\))1159 4087 y Fx(F)1204 4096 y Fm(1)1287 4072 y Fz(=3D)h(inf)7 b(\()p Fy(b)p Fz(\))1643 4087 y Fx(F)1688 4096 y Fm(0)1771 4072 y Fu(6)p Fz(=3D)44 b(0)f(in)f Fy(H)2195 4036 y Fl(1)2234 4072 y Fz(\()p Fy(G)2349 4087 y Fx(F)2394 4096 y Fm(1)2433 4072 y Fy(;)17 b Fv(F)2538 4087 y Fx(p)2577 4072 y Fz(\))42 b(and)h(\()p Fy(a)2946 4087 y Fl(2)2986 4072 y Fz(\))3024 4087 y Fx(F)3069 4096 y Fm(2)3151 4072 y Fz(=3D)i(0)d(in)456 4188 y Fy(H)545 4152 y Fl(2)584 4188 y Fz(\()p Fy(G)699 4203 y Fx(F)744 4212 y Fm(2)782 4188 y Fy(;)17 b Fv(F)887 4203 y Fx(p)926 4188 y Fz(\).)555 4406 y(Finally)36 b(assume)h(that)e(\007)d(=3D)g(0)j(and)h Fy(p)c Fz(=3D)g(2.)51 b(W)-8 b(e)36 b(set)g(\()p Fy(a)2668 4421 y Fl(1)2708 4406 y Fz(\))2746 4421 y Fx(F)2791 4430 y Fm(1)2861 4406 y Fz(=3D)d(\()p Fu(\000)p Fz(1\))3172 4421 y Fx(F)3217 4430 y Fm(1)3287 4406 y Fu(6)p Fz(=3D)g(0)456 4522 y(in)f Fy(H)658 4486 y Fl(1)697 4522 y Fz(\()p Fy(G)812 4537 y Fx(F)857 4546 y Fm(1)896 4522 y Fy(;)17 b Fv(F)1001 4537 y Fl(2)1040 4522 y Fz(\))32 b(and)h(\()p Fy(a)1389 4537 y Fl(2)1428 4522 y Fz(\))1466 4537 y Fx(F)1511 4546 y Fm(2)1577 4522 y Fz(=3D)28 b(0)k(in)h Fy(H)1965 4486 y Fl(2)2004 4522 y Fz(\()p Fy(G)2119 4537 y Fx(F)2164 4546 y Fm(2)2202 4522 y Fy(;)17 b Fv(F)2307 4537 y Fl(2)2346 4522 y Fz(\).)555 4740 y(No)m(w)41 b(b)m(y)h(\(1\))d(there)i(exists)h Fy(a)e Fu(2)g Fy(F)1901 4704 y Fw(\002)2000 4740 y Fz(suc)m(h)h(that)f (\()p Fy(a)p Fz(\))2573 4755 y Fx(F)2672 4740 y Fu(6)p Fz(=3D)g(0)f(and)h(\()p Fy(a)p Fz(\))3200 4755 y Fx(G)3255 4766 y Fs(F)3294 4782 y(i)3369 4740 y Fz(=3D)456 4867 y(\()p Fy(a)545 4882 y Fx(i)573 4867 y Fz(\))611 4882 y Fx(F)656 4892 y Fs(i)686 4867 y Fz(,)g Fy(i)d Fz(=3D)g(1,)i(2.)60 b(Since)39 b(w)m(e)g(ha)m(v)m(e)h(already)e(determined)i(that)e Fy(\030)2951 4882 y Fx(p)3028 4867 y Fu(2)f Fy(F)3208 4831 y Fw(\002)3267 4867 y Fz(,)i(w)m(e)456 4983 y(see)28 b(that)g Fy(K)34 b Fz(=3D)28 b Fy(F)14 b Fz(\()1178 4950 y Fs(p)1155 4911 y Fu(p)p 1238 4911 52 4 v 72 x Fy(a)p Fz(\))28 b(is)g(a)f(cyclic)i(extension)h(of)d(degree)h Fy(p)p Fz(.)42 b(Hence)29 b(it)e(remains)456 5099= y(to)k(sho)m(w)h (that)g(the)f(arithmetic)i(in)m(v)-5 b(arian)m(ts)32 b Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\),)31 b Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\),)31 b(and)g(\007\()p Fy(K)r(=3DF)14 b Fz(\))456 5216 y(coincide)29= b(with)g(the)f(prescrib)s(ed)i(cardinal) e(n)m(um)m(b)s(ers)i Fy(d)p Fz(,)f Fy(e)p= Fz(,)g(and)f(\007)g(resp)s (ectiv)m(ely)-8 b(.)p eop %%Page: 14 14 14 13 bop 456 255 a Fq(14)759 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456 450 y Fz(3.3.)48 b FF(Determining)38 b Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))36 b FF(and)j Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))36 b FF(via)j(annihilators.)555 654 y Fz(W)-8 b(e)31= b(\014rst)f(observ)m(e)i(some)f(relationships)h(among)e(cup)h (pro)s(ducts)f(of)g(elemen)m(ts)456 770 y(in)37 b Fy(H)663 734 y Fl(1)701 770 y Fz(\()p Fy(G)816 785 y Fx(F)875 770 y Fy(;)17 b Fv(F)980 785 y Fx(p)1019 770 y Fz(\).)56 b(Recall)37 b(that)g(Lemma)g(8)f(tells)i(us)f(that)g(if)f Fy(c)2782 785 y Fl(1)2821 770 y Fy(;)17 b(c)2907 785 y Fl(2)2981 770 y Fu(2)35 b Fy(F)3159 734 y Fw(\002)3255 770 y Fz(with)456 886 y(\()p Fy(c)536 901 y Fl(1)575 886 y Fz(\))613 901 y Fx(G)668 912 y Fs(F)707 927 y Fm(1)777 886 y Fz(=3D)28 b(\()p Fy(c)961 901 y Fl(2)1000 886 y Fz(\))1038 901 y Fx(G)1093 912 y Fs(F)1132 927 y Fm(2)1202 886 y Fz(=3D)g(0)k(then)1272 1071 y(\()p Fy(c)1352 1086 y Fl(1)1392 1071 y Fz(\))1430 1086 y Fx(F)1511 1071 y Fu([)22 b Fz(\()p Fy(c)1679 1086 y Fl(2)1718 1071 y Fz(\))1756 1086 y Fx(F)1843 1071 y Fz(=3D)27 b(0)h Fu(2)g Fy(H)2206 1030 y Fl(2)2245 1071 y Fz(\()p Fy(G)2360 1086 y Fx(F)2418 1071 y Fy(;)17 b Fv(F)2523 1086 y Fx(p)2562 1071 y Fz(\))p Fy(:)693 b Fz(\(5\))456 1238 y(Moreo)m(v)m(er,)34= b(recall)f(that)f(b)m(y)i([N,) f(Satz)f(4.1])1059 1406 y Fy(H)1148 1365 y Fl(2)1187 1406 y Fz(\()p Fy(G)1302 1421 y Fx(F)1360 1406 y Fy(;)17 b Fv(F)1465 1421 y Fx(p)1505 1406 y Fz(\))1570 1379 y Fu(\030)1571 1411 y Fz(=3D)1675 1406 y Fy(H)1764 1365 y Fl(2)1803 1406 y Fz(\()p Fy(G)1918 1421 y Fx(F)1963 1430 y Fm(1)2002 1406 y Fy(;)g Fv(F)2107 1421 y Fx(p)2146 1406 y Fz(\))22 b Fu(\002)h Fy(H)2395 1365 y Fl(2)2434 1406 y Fz(\()p Fy(G)2549 1421 y Fx(F)2594 1430 y Fm(2)2632 1406 y Fy(;)17 b Fv(F)2737 1421 y Fx(p)2776 1406 y Fz(\))p Fy(;)456 1574 y Fz(where)33= b(the)g(isomorphism)i(is)e(induced)h(b)m(y) f(the)g(restriction)h(maps)1077 1741 y(res)1197 1756 y Fx(i)1253 1741 y Fz(:)28 b Fy(H)1397 1700 y Fl(2)1435 1741 y Fz(\()p Fy(G)1550 1756 y Fx(F)1609 1741 y Fy(;)17 b Fv(F)1714 1756 y Fx(p)1753 1741 y Fz(\))28 b Fu(!)f Fy(H)2035 1700 y Fl(2)2074 1741 y Fz(\()p Fy(G)2189 1756 y Fx(F)2234 1766 y Fs(i)2264 1741 y Fy(;)17 b Fv(F)2369 1756 y Fx(p)2408 1741 y Fz(\))p Fy(;)g(i)28 b Fz(=3D)g(1)p Fy(;)17 b Fz(2)p Fy(:)456 1909 y Fz(Because)30= b(these)g(restriction)g (maps)g(comm)m(ute)g(with)g(cup)f(pro)s(duct)g(maps)h(\([Ris,)456 2025 y(Prop)s(osition)53 b(7.3,)58 b(page)53= b(191]\),)k(w)m(e)d(see)g (that)f(if)g Fy(c)2481 2040 y Fl(1)2520 2025 y Fy(;)17 b(c)2606 2040 y Fl(2)2708 2025 y Fu(2)62 b Fy(F)2913 1989 y Fw(\002)3025 2025 y Fz(suc)m(h)55 b(that)456 2141 y(\()p Fy(c)536 2156 y Fl(1)575 2141 y Fz(\))613 2156 y Fx(G)668 2167 y Fs(F)707 2182 y Fm(2)777 2141 y Fz(=3D)28 b(\()p Fy(c)961 2156 y Fl(2)1000 2141 y Fz(\))1038 2156 y Fx(G)1093 2167 y Fs(F)1132 2182 y Fm(2)1202 2141 y Fz(=3D)g(0)k(in)h Fy(H)1590 2105 y Fl(2)1629 2141 y Fz(\()p Fy(G)1744 2156 y Fx(F)1789 2165 y Fm(2)1827 2141 y Fy(;)17 b Fv(F)1932 2156 y Fx(p)1971 2141 y Fz(\))33 b(then)997 2317 y(\()p Fy(c)1077 2332 y Fl(1)1116 2317 y Fz(\))1154 2332 y Fx(F)1235 2317 y Fu([)22 b Fz(\()p Fy(c)1403 2332 y Fl(2)1442 2317 y Fz(\))1480 2332 y Fx(F)1567 2317 y Fu(6)p Fz(=3D)27 b(0)98 b(i\013)f(\()p Fy(c)2078 2332 y Fl(1)2117 2317 y Fz(\))2155 2332 y Fx(G)2210 2343 y Fs(F)2249 2358 y Fm(1)2314 2317 y Fu([)23 b Fz(\()p Fy(c)2483 2332 y Fl(2)2522 2317 y Fz(\))2560 2332 y Fx(G)2615 2343 y Fs(F)2654 2358 y Fm(1)2724 2317 y Fu(6)p Fz(=3D)28 b(0)p Fy(;)416 b Fz(\(6\))456 2496 y(where)48= b(the)g(\014rst)g(cup)h(pro)s (duct)f(lies)g(in)g Fy(H)2133 2460 y Fl(1)2172 2496 y Fz(\()p Fy(G)2287 2511 y Fx(F)2345 2496 y Fy(;)17 b Fv(F)2450 2511 y Fx(p)2490 2496 y Fz(\))47 b(and)h(the)g(second)h(cup)456 2612 y(pro)s(duct)44 b(lies)h(in)f Fy(H)1227 2576 y Fl(1)1266 2612 y Fz(\()p Fy(G)1381 2627 y Fx(F)1426 2636 y Fm(1)1464 2612 y Fy(;)17 b Fv(F)1569 2627 y Fx(p)1608 2612 y Fz(\).)78 b(By)45 b(symmetry)h(the)e(analogous)g(statemen)m(t)456 2729 y(with)33 b Fy(F)741 2744 y Fl(1)813 2729 y Fz(and)g Fy(F)1066 2744 y Fl(2)1138 2729 y= Fz(exc)m(hanged)h(holds)f(as)g(w)m (ell.)555 2932 y(No)m(w)k(w)m(e)h(calculate)g Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\).)55 b(W)-8 b(e)37= b(adopt)f(the)h(follo)m (wing)g(notation)f(for)g(an)456 3048 y(annihilator)c(of)g Fy(\036)c Fu(2)g Fy(H)1334 3012 y Fl(1)1373 3048 y Fz(\()p Fy(G)1488 3063 y Fx(F)1546 3048 y Fy(;)17 b Fv(F)1651 3063 y Fx(p)1691 3048 y Fz(\):)1075 3220 y(ann)1232 3235 y Fx(F)1277 3244 y Fm(1)1332 3220 y Fy(\036)28 b Fz(:=3D)f Fu(f)p Fy(\021)k Fu(2)e Fy(H)1861 3179 y Fl(1)1900 3220 y Fz(\()p Fy(G)2015 3235 y Fx(F)2073 3220 y Fy(;)17 b Fv(F)2178 3235 y Fx(p)2217 3220 y Fz(\))33 b Fu(j)f Fy(\021)26 b Fu([)c Fy(\036)28 b Fz(=3D)f(0)p Fu(g)p Fy(:)456 3387 y Fz(F)-8 b(or)35 b Fy(x)e Fu(2)h Fy(F)899 3351 y Fw(\002)993 3387 y Fz(w)m(e)j(ha)m(v)m(e)g Fy(x)d Fu(2)g Fy(N)1635 3403 y Fx(K)q(=3DF)1789 3387 y Fz(\()p Fy(K)1917 3351 y Fw(\002)1976 3387 y Fz(\))i(if)g(and)f(only)i(if)f(\()p Fy(x)p Fz(\))2777 3402 y Fx(F)2860 3387 y Fu([)25 b Fz(\()p Fy(a)p Fz(\))3078 3402 y Fx(F)3169 3387 y Fz(=3D)34 b(0)h(in)456 3507 y Fy(H)545 3470 y Fl(2)584 3507 y Fz(\()p Fy(G)699 3522 y Fx(F)757 3507 y Fy(;)17 b Fv(F)862 3522 y Fx(p)901 3507 y Fz(\).)60 b(Therefore,)41= b(using)f(\(5\))e(and)g(\(6\))f(with)i (the)f(fact)g(that)g(in)g(ev)m(ery)456 3623 y(case)33 b(\()p Fy(a)p Fz(\))789 3638 y Fx(G)844 3649 y Fs(F)883 3664 y Fm(2)953 3623 y Fz(=3D)28 b(0,)698 3805 y Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)h(dim)1323 3820 y Ft(F)1364 3828 y Fs(p)1421 3805 y Fy(N)10 b Fz(\()p Fy(K)1637 3763 y Fw(\002)1696 3805 y Fz(\))p Fy(=3DF)1860 3763 y Fw(\002)p Fx(p)1057 3965 y Fz(=3D)28 b(dim)1323 3980 y Ft(F)1364 3988 y Fs(p)1404 3965 y Fu(f)p Fz(\()p Fy(x)p Fz(\))1585 3980 y Fx(F)1672 3965 y Fu(2)g Fy(H)1855 3924 y Fl(1)1894 3965 y Fz(\()p Fy(G)2009 3980 y Fx(F)2067 3965 y Fy(;)17 b Fv(F)2172 3980 y Fx(p)2211 3965 y Fz(\))33 b Fu(j)f Fz(\()p Fy(a)p Fz(\))2469 3980 y Fx(F)2550 3965 y Fu([)22 b Fz(\()p Fy(x)p Fz(\))2769 3980 y Fx(F)2856 3965 y Fz(=3D)28 b(0)p Fu(g)1057 4125 y Fz(=3D)g(dim)1323 4140 y Ft(F)1364 4148 y Fs(p)1404 4125 y Fu(f)p Fz(\()p Fy(y)t Fz(\))1582 4140 y Fx(F)1627 4149 y Fm(1)1692 4125 y Fu(2)g Fy(H)1875 4084 y Fl(1)1914 4125 y Fz(\()p Fy(G)2029 4140 y Fx(F)2074 4149 y Fm(1)2112 4125 y Fy(;)17 b Fv(F)2217 4140 y Fx(p)2257 4125 y Fz(\))32 b Fu(j)g Fz(\()p Fy(a)p Fz(\))2514 4140 y Fx(G)2569 4151 y Fs(F)2608 4166 y Fm(1)2673 4125 y Fu([)23 b Fz(\()p Fy(y)t Fz(\))2890 4140 y Fx(F)2935 4149 y Fm(1)3000 4125 y Fz(=3D)k(0)p Fu(g)1193 4288 y Fz(+)22 b(dim)1453 4303 y Ft(F)1494 4311 y Fs(p)1551 4288 y Fy(H)1640 4246 y Fl(1)1679 4288 y Fz(\()p Fy(G)1794 4303 y Fx(F)1839 4312 y Fm(2)1877 4288 y Fy(;)17 b Fv(F)1982 4303 y Fx(p)2021 4288 y Fz(\))1057 4451 y(=3D)28 b(dim)1323 4466 y Ft(F)1364 4474 y Fs(p)1421 4451 y Fz(ann)1578 4466 y Fx(F)1623 4475 y Fm(1)1661 4451 y Fz(\()p Fy(a)p Fz(\))1788 4466 y Fx(G)1843 4477 y Fs(F)1882 4492 y Fm(1)1947 4451 y Fz(+)22 b(dim)2208 4466 y Ft(F)2249 4474 y Fs(p)2305 4451 y Fy(F)2382 4410 y Fw(\002)2368 4476 y Fl(2)2441 4451 y Fy(=3DF)2567 4404 y Fw(\002)p Fx(p)2553 4476 y Fl(2)2661 4451 y Fy(:)555 4718 y Fz(If)52 b(\007)59 b(=3D)h(1)51 b(then)h Fy(H)1373 4682 y Fl(1)1412 4718 y Fz(\()p Fy(G)1527 4733 y Fx(F)1572 4742 y Fm(0)1611 4718 y Fy(;)17 b Fv(F)1716 4733 y Fx(p)1755 4718 y Fz(\))60 b(=3D)f(0,)e(and)51= b(so)h(b)m(y)g (Lemma)g(4)f(w)m(e)i(ha)m(v)m(e)456 4834 y(dim)618 4849 y Ft(F)659 4857 y Fs(p)716 4834 y Fz(ann)873 4849 y Fx(F)918 4858 y Fm(1)956 4834 y Fz(\()p Fy(a)p Fz(\))1083 4849 y Fx(G)1138 4860 y Fs(F)1177 4875 y Fm(1)1247 4834 y Fz(=3D)28 b(1.)555 5040 y(If)33 b Fy(p)27 b(>)h= Fz(2)k(and)h(\007)27 b(=3D)h(0)k(then)h(b)m(y)h(Lemma)f(4,)1138 5208 y(dim)1301 5223 y Ft(F)1342 5231 y Fs(p)1398 5208 y Fz(ann)1555 5223 y Fx(F)1600 5232 y Fm(1)1639 5208 y Fz(\()p Fy(a)p Fz(\))1766 5223 y Fx(G)1821 5234 y Fs(F)1860 5249 y Fm(1)1930 5208 y Fz(=3D)27 b(dim)2196 5223 y Ft(F)2237 5231 y Fs(p)2294 5208 y Fz(ann)2451 5223 y Fx(F)2496 5232 y Fm(0)2534 5208 y Fz(\()p Fy(b)p Fz(\))2651 5223 y Fx(F)2696 5232 y Fm(0)2735 5208 y Fy(;)p eop %%Page: 15 15 15 14 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)292 b(15)456 460 y Fz(where)24 b Fy(b)g Fz(w)m(as)g(c)m(hosen)h(so)f(that)f Fy(K)1670 475 y Fl(0)1737 460 y Fz(=3D)28 b Fy(F)1904 475 y Fl(0)1943 460 y Fz(\()2009 411 y Fs(p)1985 375 y Fu(p)p 2069 375 42 4 v 2069 460 a Fy(b)p Fz(\))23= b(w)m(as)i(the)e(\014xed)i(\014eld)f (of)f(the)h(factor)456 576 y Fv(Z)522 591 y Fx(p)593 576 y Fz(in)32 b Fy(G)783 540 y Fx(ab)783 601 y(F)828 610 y Fm(0)866 576 y Fz(.)43 b(No)m(w)32 b(b)s(ecause)h(\()p Fy(b)p Fz(\))1634 591 y Fx(F)1679 600 y Fm(0)1738 576 y Fu([)20 b Fz(\()p Fy(b)p Fz(\))1941 591 y Fx(F)1986 600 y Fm(0)2052 576 y Fz(=3D)28= b(0)j(as)g(an)h(iden)m(tit)m(y)h(in)f Fy(H)3050 540 y Fl(2)3089 576 y Fz(\()p Fy(G)3204 591 y Fx(F)3262 576 y Fy(;)17 b Fv(F)3367 591 y Fx(p)3406 576 y Fz(\))456 692 y(for)32 b Fy(p)27 b(>)h Fz(2,)1404 811 y(1)g Fu(\024)g Fz(dim)1748 826 y Ft(F)1789 834 y Fs(p)1846 811 y Fz(ann)2003 826 y Fx(F)2048 835 y Fm(0)2086 811 y Fz(\()p Fy(b)p Fz(\))2203 826 y Fx(F)2248 835 y Fm(0)2315 811 y Fu(\024)g Fz(2)p Fy(:)456 954 y= Fz(On)44 b(the)h(other)f(hand,)k(let)d Fy(c)i Fu(2)h Fy(F)1795 918 y Fw(\002)1884 954 y Fu(n)30 b Fy(F)2041 918 y Fw(\002)p Fx(p)2180 954 y Fz(b)s(e)44 b(de\014ned)i(so)f(that)f Fy(F)3090 969 y Fl(0)3129 954 y Fz(\()3195 920 y Fs(p)3171 882 y Fu(p)p 3255 882 V 3255 954 a Fy(c)p Fz(\))g(is)456 1070 y(con)m(tained)d(in)f(the)g(\014xed)h(\014eld)g(of)e(the)h(factor) g Fv(Z)p Fy(=3Dp)p Fv(Z)g Fz(in)g Fy(G)2711 1034 y Fx(ab)2711 1095 y(F)2756 1104 y Fm(0)2794 1070 y Fz(.)66= b(Then)41 b(b)m(y)h(\(2\))o(,)456 1192 y Fu(f)p Fz(\()p Fy(b)p Fz(\))623 1207 y Fx(F)668 1216 y Fm(0)706 1192 y Fy(;)17 b Fz(\()p Fy(c)p Fz(\))868 1207 y Fx(F)913 1216 y Fm(0)951 1192 y Fu(g)31 b Fz(spans)g Fy(H)1385 1155 y Fl(1)1424 1192 y Fz(\()p Fy(G)1539 1207 y Fx(F)1584 1216 y Fm(0)1623 1192 y Fy(;)17 b Fv(F)1728 1207 y Fx(p)1767 1192 y Fz(\).)43 b(F)-8 b(urthermore,)31 b(since)h(b)m(y)g(Lemma)f(5,)g Fy(H)3370 1208 y Fx(p)3406 1189 y Fm(3)456 1308 y= Fz(is)i(not)g(a)g (quotien)m(t)h(of)f Fy(G)1384 1323 y Fx(F)1429 1332 y Fm(0)p Fs(;p)1518 1308 y Fz(,)g(\()p Fy(b)p Fz(\))1695 1323 y Fx(F)1740 1332 y Fm(0)1802 1308 y Fu([)23 b Fz(\()p Fy(c)p Fz(\))2009 1323 y Fx(F)2054 1332 y Fm(0)2120 1308 y Fu(6)p Fz(=3D)29 b(0)k(b)m(y)h([M,)f(Corollary)-8 b(,)34 b(page)f(523)456 1425 y(and)f(Theorem)i(3\(A\)].)f(W)-8 b(e)33 b(conclude)h(that)e(dim)2285 1440 y Ft(F)2326 1448 y Fs(p)2382 1425 y Fz(ann)2539 1440 y Fx(F)2584 1449 y Fm(0)2623 1425 y Fz(\()p Fy(b)p Fz(\))2740 1440 y Fx(F)2785 1449 y Fm(0)2851 1425 y Fz(=3D)c(1.)555 1625 y(Finally)-8 b(,)33 b(if)g Fy(p)27 b= Fz(=3D)h(2)k(and)h(\007)27 b(=3D)h(0)k(then)h(again)f(b)m(y)i(Lemma)f(4)1097 1784 y(dim)1260 1799 y Ft(F)1301 1808 y Fm(2)1356 1784 y Fz(ann)1513 1799 y Fx(F)1558 1808 y Fm(1)1596 1784 y Fz(\()p Fy(a)p Fz(\))1723 1799 y Fx(G)1778 1810 y Fs(F)1817 1825 y Fm(1)1888 1784 y Fz(=3D)27 b(dim)2154 1799 y Ft(F)2195 1808 y Fm(2)2250 1784 y Fz(ann)2407 1799 y Fx(F)2452 1808 y Fm(0)2490 1784 y Fz(\()p Fu(\000)p Fz(1\))2692 1799 y Fx(F)2737 1808 y Fm(0)2776 1784 y Fy(:)456 1960 y Fz(As)49 b Fy(F)679 1975 y Fl(0)773 1960 y Fz(=3D)54 b Fv(R)p Fz(,)f(dim)1217 1975 y Ft(F)1258 1984 y Fm(2)1313 1960 y Fy(F)1390 1918 y Fw(\002)1376 1984 y Fl(0)1449 1960 y Fy(=3DF)1575 1918 y Fw(\002)p Fl(2)1561 1984 y(0)1724 1960 y Fz(=3D)h(1)48 b(and)h(\()p Fu(\000)p Fz(1\))2359 1975 y Ft(R)2444 1960 y Fu([)33 b Fz(\()p Fu(\000)p Fz(1\))2745 1975 y Ft(R)2852 1960 y Fu(6)p Fz(=3D)55 b(0,)d(yielding)456 2076 y(dim)618 2091 y Ft(F)659 2100 y Fm(2)714 2076 y Fz(ann)871 2091 y Fx(F)916 2100 y Fm(1)955 2076 y Fz(\()p Fy(a)p Fz(\))1082 2091 y Fx(G)1137 2102 y Fs(F)1176 2117 y Fm(1)1246 2076 y Fz(=3D)27 b(0)p Fy(:)555 2284 y Fz(Com)m(bining)34= b(our)f(results)h (with)f(\(4\))o(,)g(w)m(e)h(ha)m(v)m(e)g(that)e Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g Fy(e)p Fz(.)555 2484 y(No)m(w)42= b(w)m(e)g(turn)f(to)f(a)h(similar)h(calculation)f(of)g Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\).)67 b(Again)41= b(using)i(\(5\))456 2600 y(and)32= b(\(6\))g(with)i(the)f(fact)f(that)g(in)h(ev)m(ery)i (case)e(\()p Fy(a)p Fz(\))2301 2615 y Fx(G)2356 2626 y Fs(F)2395 2641 y Fm(2)2465 2600 y Fz(=3D)28 b(0,)1017 2776 y Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(dim)1648 2791 y Ft(F)1689 2799 y Fs(p)1729 2776 y Fz(\()p Fy(H)1856 2735 y Fl(1)1895 2776 y Fz(\()p Fy(G)2010 2791 y Fx(F)2068 2776 y Fy(;)17 b Fv(F)2173 2791 y Fx(p)2212 2776 y Fz(\))p Fy(=3D)g Fz(ann)2473 2791 y Fx(F)2532 2776 y Fz(\()p Fy(a)p Fz(\))2659 2791 y Fx(F)2717 2776 y Fz(\))1382 2936 y(=3D)27 b(dim)1648 2951 y Ft(F)1689 2959 y Fs(p)1729 2936 y Fz(\()p Fy(H)1856 2895 y Fl(1)1895 2936 y Fz(\()p Fy(G)2010 2951 y Fx(F)2055 2960 y Fm(1)2093 2936 y Fy(;)17 b Fv(F)2198 2951 y Fx(p)2237 2936 y Fz(\))p Fy(=3D)g Fz(ann)2498 2951 y Fx(F)2543 2960 y Fm(1)2581 2936 y Fz(\()p Fy(a)p Fz(\))2708 2951 y Fx(G)2763 2962 y Fs(F)2802 2977 y Fm(1)2845 2936 y Fz(\))1382 3084 y(=3D)27 b Fy(d:)456 3248 y Fz(F)-8= b(or)37 b(this)i(last)f(equalit)m(y)i(w)m(e)f(use)g(\(3\))f(together)g(with)h (calculations)g(of)f(the)h(di-)456 3365 y(mension)k(of)f(ann)1123 3380 y Fx(F)1168 3389 y Fm(1)1206 3365 y Fz(\()p Fy(a)p Fz(\))1333 3380 y Fx(G)1388 3391 y Fs(F)1427 3406 y Fm(1)1512 3365 y Fz(already)h(ac)m(hiev)m(ed.)74 b(Hence)44 b Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))42 b(=3D)i Fy(d)e Fz(in)g(all)456 3481 y(cases.)456 3743 y(3.4.)48 b FF(Determining)38 b Fz(\007)g FF(via)g(quotien)m(ts)f(of)h Fy(G)2295 3758 y Fx(F)2353 3743 y FF(.)555 3943 y Fz(It)26= b(remains)g(to)f(sho)m(w)i (that)e(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(\007.)41 b(First)26 b(consider)g(the)g(case)g(\007)i(=3D)g(1.)456 4059 y(Since)i Fy(F)770 4074 y Fl(0)837 4059 y Fz(=3D)e Fv(C)p Fz(,)i Fy(\030)1113 4074 y Fx(p)1181 4059 y Fz(is)g(a)f Fy(p)p Fz(th)h(p)s(o)m(w)m(er)g(in)g Fy(F)1980 4074 y Fl(1)2019 4059 y Fz(,)g(and)g Fy(F)2326 4074 y Fl(1)2365 4059 y Fz(\()2431 4026 y Fs(p)2407 3987 y Fu(p)p 2491 3987 52 4 v 2491 4059 a Fy(a)p Fz(\))f(em)m(b)s(eds)i(in)f(a)f Fv(Z)p Fy(=3Dp)3306 4023 y Fl(2)3345 4059 y Fv(Z)p Fz(-)456 4182 y(extension)50 b Fy(F)966 4197 y Fl(1)1006 4182 y Fz(\()1071 4148 y Fs(p)1103 4127 y Fm(2)1083 4110 y Fu(p)p 1166 4110 V 72 x Fy(a)p Fz(\))f(of)f Fy(F)1494 4197 y Fl(1)1534 4182 y Fz(.)92 b(Then)50 b(the)f(surjection)h Fy(G)2649 4197 y Fx(F)2694 4206 y Fm(1)2787 4182 y Fu(!)55 b Fz(Gal\()p Fy(K)3216 4197 y Fl(1)3255 4182 y Fy(=3DF)3367 4197 y Fl(1)3406 4182 y Fz(\))456 4298 y(factors)44 b(through)h Fv(Z)p Fy(=3Dp)1330 4262 y Fl(2)1370 4298 y Fv(Z)p Fz(.)81 b(F)-8 b(ollo)m(wing)45 b(the)g(surjection)h(with)f(the)h(canonical)456 4414 y(surjection)h Fy(G)994 4429 y Fx(F)1105 4414 y Fu(!)k Fy(G)1333 4429 y Fx(F)1378 4438 y Fm(1)1416 4414 y Fz(,)g(w)m(e)c(see)h(that)f Fv(Z)p Fy(=3Dp)2213 4378 y Fl(2)2252 4414 y Fv(Z)g Fz(is)g(a)g(factor)f(group)h(of)f Fy(G)3359 4429 y Fx(F)3417 4414 y Fz(.)456 4530= y(Moreo)m(v)m(er,)34 b(b)m(y)f(Lemma)h(7,)e(the)h(surjection)h Fy(G)2196 4545 y Fx(F)2282 4530 y Fu(!)28 b Fv(Z)p Fy(=3Dp)p Fv(Z)33= b Fz(corresp)s(onds)h(to)e Fy(K)7 b Fz(.)456 4647 y(Hence)41 b Fy(K)r(=3DF)52 b Fz(em)m(b)s(eds)42 b(in)e(a)f Fv(Z)p Fy(=3Dp)1730 4611 y Fl(2)1770 4647 y Fv(Z)p Fz(-extension)i(of)e Fy(F)14 b Fz(.)65 b(By)40 b([A,)h(Theorem)g(3],)456 4763 y Fy(\030)499 4778 y Fx(p)566 4763 y Fu(2)28 b Fy(N)738 4778 y Fx(K)q(=3DF)892 4763 y Fz(\()p Fy(K)1020 4727 y Fw(\002)1079 4763 y Fz(\).)43 b(Therefore)34 b(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(1)h(=3D)f(\007.)555 4980 y(No)m(w)43 b(consider)h(the)f(case)h(\007)g(=3D)h(0)d(and)g Fy(p)j(>)f Fz(2.)74 b(Because)44 b Fy(K)2933 4995 y Fl(0)3017 4980 y Fz(=3D)g Fy(F)3200 4995 y Fl(0)3240 4980 y Fz(\()3305 4931 y Fs(p)3282 4895 y Fu(p)p 3365 4895 42 4 v 85 x Fy(b)p Fz(\))456 5096 y(do)s(es)c(not)h(em)m(b)s(ed)g(in)g(a)f Fv(Z)p Fy(=3Dp)1556 5060 y Fl(2)1596 5096 y Fv(Z)p= Fz(-extension)i(of)e Fy(F)2316 5111 y Fl(0)2356 5096 y Fz(,)i Fy(\030)2468 5111 y Fx(p)2560 5096 y Fy(=3D)-61 b Fu(2)42 b Fy(N)2734 5112 y Fx(K)2794 5121 y Fm(0)2828 5112 y Fx(=3DF)2908 5121 y Fm(0)2947 5096 y Fz(\()p Fy(K)3075 5055 y Fw(\002)3068 5121 y Fl(0)3134 5096 y Fz(\).)67 b(\(See)456 5216= y(3.2)34 b(for)g(the)h(de\014nition)h(of)e Fy(K)1568 5231 y Fl(0)1608 5216 y Fz(.\))50 b(Hence)36 b(\()p Fy(b)p Fz(\))2132 5231 y Fx(F)2177 5240 y Fm(0)2239 5216 y Fu([)24 b Fz(\()p Fy(\030)2410 5231 y Fx(p)2449 5216 y Fz(\))2487 5231 y Fx(F)2532 5240 y Fm(0)2602 5216 y Fu(6)p Fz(=3D)31 b(0)k(in)g Fy(H)2998 5179 y Fl(1)3037 5216 y Fz(\()p Fy(G)3152 5231 y Fx(F)3197 5240 y Fm(0)3235 5216 y Fy(;)17 b Fv(F)3340 5231 y Fx(p)3379 5216 y Fz(\),)p eop %%Page: 16 16 16 15 bop 456 255 a Fq(16)759 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456 450 y Fz(and,)32 b(b)m(y)g(Lemma)g(4,)g(\(inf)6 b(\()p Fy(b)p Fz(\)\))21 b Fu([)f Fz(\()p Fy(\030)1759 465 y Fx(p)1798 450 y Fz(\))1836 465 y Fx(F)1881 474 y Fm(1)1947 450 y Fu(6)p Fz(=3D)28 b(0)j(in)h Fy(H)2333 414 y Fl(1)2372 450 y Fz(\()p Fy(G)2487 465 y Fx(F)2532 474 y Fm(1)2570 450 y Fy(;)17 b Fv(F)2675 465 y Fx(p)2714 450 y Fz(\))32 b(as)f(w)m(ell.)45 b(Cho)s(ose)456 572 y Fy(b)497 587 y Fl(1)574 572 y Fu(2)37 b Fy(F)754 530 y Fw(\002)740 596 y Fl(1)851 572 y Fz(so)h(that)g(\()p Fy(b)1272 587 y Fl(1)1312 572 y Fz(\))1350 587 y Fx(F)1395 596 y Fm(1)1470 572 y Fz(=3D)f(inf)7 b(\()p Fy(b)p Fz(\))1818 587 y Fx(F)1863 596 y Fm(0)1940 572 y Fz(and)38 b(set)h Fy(K)2376 587 y Fl(1)2452 572 y Fz(=3D)e Fy(F)2628 587 y Fl(1)2667 572 y Fz(\()2733 531 y Fs(p)2709 494 y Fu(p)p 2793 494 81 4 v 2793 572 a Fy(b)2834 587 y Fl(1)2874 572 y Fz(\).)59 b(Then)40 b Fy(\030)3302 587 y Fx(p)3390 572 y Fy(=3D)-61 b Fu(2)456 689 y Fy(N)534 705 y Fx(K)594 714 y Fm(1)628 705 y Fx(=3DF)708 714 y Fm(1)747 689 y Fz(\()p Fy(K)875 648 y Fw(\002)868 714 y Fl(1)934 689 y Fz(\))28= b(and)h(b)m(y)g([A)q(,) g(Theorem)h(3])e(the)h(\014eld)h(extension)g Fy(K)2868 704 y Fl(1)2908 689 y Fy(=3DF)3020 704 y Fl(1)3087 689 y Fz(do)s(es)f(not)456 809 y(em)m(b)s(ed)43 b(in)f(a)g Fv(Z)p Fy(=3Dp)1152 772 y Fl(2)1192 809 y Fv(Z)p= Fz(-extension)i(of)e Fy(F)1916 824 y Fl(1)1955 809 y Fz(.)72 b(Therefore)44 b(the)e(surjection)h Fy(G)3217 824 y Fx(F)3262 833 y Fm(1)3345 809 y Fu(!)456 935 y Fz(Gal)o(\()p Fy(F)709 950 y Fl(1)749 935 y Fz(\()814 885 y Fs(p)791 850 y Fu(p)p 874 850 42 4 v 85 x Fy(b)p Fz(\))p Fy(=3DF)1065 950 y Fl(1)1105 935 y Fz(\))33 b(do)s(es)g(not)g(factor)g(through)g Fv(Z)p Fy(=3Dp)2383 898 y Fl(2)2423 935 y Fv(Z)p Fz(.)45 b(F)-8 b(rom)33 b(Lemmas)h(7)f(and)456 1051 y(9)46= b(w)m(e)i(see)g(that)f (surjection)h Fy(G)1646 1066 y Fx(F)1756 1051 y Fu(!)k Fz(Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\))46= b(do)s(es)h(not)g(factor)f (through)456 1167 y Fv(Z)p Fy(=3Dp)620 1131 y Fl(2)659 1167 y Fv(Z)p Fz(.)71 b(Again)41= b(b)m(y)i([A,)h(Theorem)e(3])g(w)m(e)g (conclude)h(that)f Fy(\030)2796 1182 y Fx(p)2889 1167 y Fy(=3D)-60 b Fu(2)43 b Fy(N)3065 1183 y Fx(K)q(=3DF)3219 1167 y Fz(\()p Fy(K)3347 1131 y Fw(\002)3406 1167 y Fz(\))456 1283 y(and)32 b(therefore)h(\007\()p Fy(K)r(=3DF)14 b= Fz(\))27 b(=3D)h(0)f(=3D)h(\007.)555 1480 y(Finally)35= b(consider)h(the)f(case)h (\007)31 b(=3D)g(0)k(and)g Fy(p)c Fz(=3D)g(2.)50 b(Because)36 b Fu(\000)p Fz(1)43 b Fy(=3D)-61 b Fu(2)32 b Fy(N)3161 1496 y Ft(C)p Fx(=3D)p Ft(R)3297 1480 y Fz(\()p Fv(C)p Fz(\))456 1602 y(from)25 b(Lemma)i(4)e(w)m(e)i(conclude)g(that)f Fu(\000)p Fz(1)39 b Fy(=3D)-60 b Fu(2)28 b Fy(N)2157 1617 y Fx(K)2217 1626 y Fm(1)2251 1617 y Fx(=3DF)2331 1626 y Fm(1)2370 1602 y Fz(\()p Fy(K)2498 1561 y Fw(\002)2491 1626 y Fl(1)2557 1602 y Fz(\).)41 b(By)26 b([A,)h(Theorem)h(3])456 1718 y(the)38 b(surjection)h Fy(G)1159 1733 y Fx(F)1204 1742 y Fm(1)1279 1718 y Fu(!)d Fz(Gal)o(\()p Fy(K)1688 1733 y Fl(1)1727 1718 y Fy(=3DF)1839 1733 y Fl(1)1879 1718 y Fz(\))h(do)s(es)i(not)e(factor)h(through)f Fv(Z)p Fy(=3D)p Fz(4)p Fv(Z)p Fz(.)60 b(As)456 1834 y(in)39= b(the)h(previous)g(case,)i (b)m(y)e(Lemmas)h(7)e(and)g(9)g(w)m(e)h(see)g(that)f Fy(K)r(=3DF)53 b Fz(do)s(es)39 b(not)456 1950 y(em)m(b)s(ed)30 b(in)e(a)h Fv(Z)p Fy(=3D)p Fz(4)p Fv(Z)p Fz(-extension)h(of)e Fy(F)14 b Fz(.)42 b(Again)29= b(b)m(y)g([A,)h(Theorem)g(3])e(w)m(e)i (see)g(that)456 2067 y Fu(\000)p Fz(1)39 b Fy(=3D)-60 b Fu(2)28 b Fy(N)782 2082 y Fx(K)q(=3DF)936 2067 y Fz(\()p Fy(K)1064 2031 y Fw(\002)1123 2067 y Fz(\))33 b(and)f(therefore)h (\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)h(0)f(=3D)h(\007.)555 2263= y(Hence)i(w)m(e)f(ha)m(v)m(e)h(c)m(hec)m(k)m(ed)h(in)e(all)f (cases)h(that)g(\007\()p Fy(K)r(=3DF)14 b Fz(\))26= b(=3D)i(\007,)h(and)f (our)g(pro)s(of)456 2380 y(of)k(Theorem)i(1)e(is)h(no)m(w)g(complete.) 1601 b Fi(\003)456 2550 y FF(Remark.)45 b= Fz(In)c([EH,)h(Lemma)e(1.2)g (and)g(Prop)s(osition)g(1.3])f(Efrat)h(and)g(Haran)456 2667= y(construct)c(some)g(\014elds)h(with)f(prescrib)s(ed)h(absolute)f (Galois)f(groups)h(together)456 2783= y(with)e(some)g(b)s(ounds)g(on)f (the)h(transcendence)i(degrees)f(of)e(these)h(\014elds.)48 b(These)456 2899 y(b)s(ounds,)33= b(together)g(with)g(the)g(replacemen)m (t)i(of)d Fv(C)g Fz(b)m(y)i(an)e(algebraic)h(closure)h(of)456 3015 y Fv(Q)d Fz(and)h(of)f Fv(R)g= Fz(b)m(y)h(a)f(real-closed)i (algebraic)e(n)m(um)m(b)s(er)i(\014eld)f Fy(R)h= Fz(in)e(the)h(case)g (when)456 3132 y Fy(p)27 b Fz(=3D)h(2)k(and)h(\007)27= b(=3D)h(0)k(in)h(our) f(pro)s(of)g(ab)s(o)m(v)m(e,)i(yield)f(the)g(follo)m(wing)g(prop)s (osition.)456 3302 y FF(Prop)s(osition)42 b(1.)h Fp(Supp)-5 b(ose)38 b(that)h Fy(p)f Fp(is)g(a)g(prime)g(numb)-5 b(er,)39 b(and)e(let)i Fy(d)p Fp(,)g Fy(e)p Fp(,)g(and)456 3419 y Fz(\007)27 b Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)34 b Fp(b)-5 b(e)34 b(c)-5 b(ar)g(dinal)33 b(numb)-5 b(ers)34 b(such)g(that)h(if)f Fz(\007)28 b(=3D)f(0)34 b Fp(then)h Fz(1)27 b Fu(\024)h Fy(d)p Fp(,)34 b(if)g Fy(p)28 b(>)g Fz(2)456 3535 y Fp(then)38 b Fz(1)e Fu(\024)g Fy(e)p Fp(,)k(and)e(if)h Fy(p)c Fz(=3D)h(2)i Fp(and)h Fz(\007)c(=3D)g(1)k Fp(then)g Fz(1)c Fu(\024)h Fy(e)p Fp(.)58 b(Then)38 b(ther)-5 b(e)39 b(exists)g(a)456 3651 y(\014eld)g Fy(F)54 b Fp(c)-5 b(ontaining)39 b Fv(Q)p Fz(\()p Fy(\030)1426 3666 y Fx(p)1465 3651 y Fz(\))h Fp(and)g(a)g(cyclic)f(Galois)h(extension)f Fy(K)47 b Fp(of)40 b(de)-5 b(gr)g(e)g(e)39 b Fy(p)456 3767 y Fp(over)34 b Fy(F)49 b Fp(such)34 b(that)851 3924 y Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g Fy(e;)117 b(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))26 b(=3D)i Fy(d;)116 b Fp(and)134 b Fz(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(\007)p Fy(:)456 4081 y Fp(Mor)-5 b(e)g(over)1220 4197 y Fz(tr)p Fy(:)17 b Fz(deg)r(\()p Fy(F)8 b(=3D)p Fv(Q)p Fz(\))28 b Fu(\024)g Fz(1)22 b(+)g(max)p Fu(f)p Fy(e;)17 b(d)22 b Fz(+)g(1)p Fu(g)p Fy(:)456 4334 y Fp(In)41 b(p)-5 b(articular)41 b(if)h Fy(d;)17 b(e)39 b Fu(2)i Fv(N)27 b Fu([)h(f)p Fz(0)p Fu(g)p Fp(,)42 b(ther)-5 b(e)42= b(exists)f(a)h(Galois)f(cyclic)g (extension)456 4450 y Fy(K)r(=3DF)50 b Fp(of)37 b(de)-5 b(gr)g(e)g(e)36 b Fy(p)h Fp(with)g(pr)-5 b(escrib)g(e)g(d)37 b(invariants)f Fy(d)p Fp(,)h Fy(e)p Fp(,)h(and)f= Fz(\007)g Fp(of)g(\014nite)f(tr)-5 b(an-)456 4566 y(sc)g(endenc)g(e)33 b(de)-5 b(gr)g(e)g(e)34 b(over)h(its)g(prime)f(\014eld)g Fv(Q)p Fp(.)1393 4845 y Fz(4.)48 b Fj(A)m(ckno)n(wledgements)555 5099 y Fz(W)-8 b(e)46= b(w)m(ould)g(lik)m(e)g(to)e(thank)i(the)f (organizers)h(of)e(the)i(MSRI)f(programs)g(on)456 5216 y(Galois)26= b(theory)i(and)f(the)g(MSRI)g(sta\013)g(for)f(giving)i(us)f (the)g(opp)s(ortunit)m(y)h(to)e(meet)p eop %%Page: 17 17 17 16 bop 826 251 a Fq(GALOIS)33= b(MODULE)f(CONSTR)n(UCTION)h(AND)f (CLASSIFICA)-6 b(TION)292 b(17)456 450 y Fz(and)28= b(to)h(b)s(egin)g (our)f(collab)s(oration)g(in)h(the)g(F)-8 b(all)29 b(of)f(1999.)41 b(W)-8 b(e)29 b(are)g(also)g(grateful)456 566 y(to)24 b(A.)h(W)-8 b(adsw)m(orth)26= b(for)f(stim)m(ulating)h(con)m(v)m (ersations.)43 b(The)26 b(\014rst)g(author)e(is)i(v)m(ery)456 683 y(appreciativ)m(e)35= b(of)f(the)h(kind)g(assistance)h(of)d(Ron)h (Hemphill,)i(manager)e(of)g(Iv)m(est)456 799= y(Prop)s(erties)44 b(Limited)g(\(London,)i(Canada\),)g(for)c(ha)m(ving)i(pro)m(vided)h (excellen)m(t)456 915 y(w)m(orking)33 b(conditions.)1660 1950 y Fj(References)456 2227 y FE([A])133 b(A.)28= b(Alb)r(ert.)g Fr(On)h(cyclic)i(\014elds)p FE(.)d(T)-7 b(rans.)27= b(Amer.)h(Math.)f (So)r(c.)h Fc(37)f FE(\(1935\),)f(454{462.)456 2326 y([A)-7 b(T])80 b(E.)26 b(Artin)g(and)h(J.)f(T)-7 b(ate.)26= b Fr(Class)j(\014eld)h(the)l(ory)p= FE(.)d(Second)f(edition.)g(Redw)n(o)r (o)r(d)g(Cit)n(y)-7 b(,)27 b(CA:)697 2426 y(Addison-W)-7 b(esley)27 b(Adv)-5 b(anced)28 b(Bo)r(ok)e(Program,)f(1990.)456 2526 y([E1])96 b(I.)22 b(Efrat.)g Fr(Pr)l(o-)p= FB(p)p Fr(-Galois)k(gr)l(oups)f(of)h(algebr)l(aic)h(extensions)d(of)i Fb(Q)p FE(.)c(J.)g(Num)n(b)r(er)h(Theory)697 2625 y= Fc(64)k FE(\(1997\),)f(84{99.)456 2725 y([E2])p 697 2725 250 4 v 359 w(.)38 b Fr(Finitely)j(gener)l(ate)l(d)g(pr)l(o-)p FB(p)p Fr(-absolute)f(Galois)i(gr)l(oups)e(over)h(glob)l(al)g(\014elds) p FE(.)697 2825 y(J.)27 b(Num)n(b)r(er)h(Theory)f= Fc(77)h FE(\(1999\),)e(83{96.)456 2924 y([EH])76 b(I.)38= b(Efrat)f(and)h(D.)h (Haran.)f= Fr(On)g(Galois)k(gr)l(oups)d(over)i(pythagor)l(e)l(an)g(and)f (semi-r)l(e)l(al)697 3024 y(close)l(d)30= b(\014elds)p FE(.)f(Israel)d(J.)h(Math.)h Fc(85)f FE(\(1994\),)g(no.)g(1-3,)g (57{78.)456 3123 y([JP])95 b(C.)22 b(U.)h(Jensen)g(and)f(A.)h(Prestel.) f= Fr(R)l(e)l(alization)k(of)g(\014nitely)f(gener)l(ate)l(d)h(pr)l (o\014nite)f(gr)l(oups)697 3223 y(by)34= b(maximal)h(ab)l(elian)g (extensions)e(of)i(\014elds)p FE(.)d(J.)g(reine)g(angew.)f(Math.)h Fc(447)g FE(\(1994\),)697 3323 y(201{218.)456 3422 y([K])130 b(J.)63 b(Ko)r(enigsmann.)g= Fr(Solvable)h(absolute)g(Galois)g(gr)l (oups)f(ar)l(e)h(metab)l(elian)p FE(.)h(In-)697 3522 y(v)n(en)n(t.)27 b(Math.)h Fc(144)e FE(\(2001\),)h(1{22.)456 3622 y([La])101 b(S.)20 b(Lang.)g= Fr(A)n(lgebr)l(a)p FE(.)h(Revised)f(third)h(edition.)f(Graduate)g(T)-7= b(exts)20 b(in)h(Mathematics)f Fc(211)p FE(.)697 3721 y(New)27 b(Y)-7 b(ork:)37 b(Springer-V)-7 b(erlag,)25 b(2002.)456 3821 y([M])119 b(R.)44 b(Massy)-7 b(.)42 b Fr(Construction)j(de)g FB(p)p Fr(-extensions)f(Galoisiennes)i(d'un)f(c)l(orps)g(de)g(c)l(ar-) 697 3920 y(act)n(\023)-40 b(eristique)30 b(di\013)n(\023)-40 b(er)l(ente)30 b(de)h FB(p)p FE(.)c(J.)g(Algebra)g= Fc(109)g FE(\(1987\),)f(no.)i(2,)f(508{535.)456 4020 y([MS])73 b(J.)36 b(Min\023)-42 b(a)n(\024)j(c)36= b(and)h(J.)f(Sw)n(allo)n(w.)g Fr(Galois)k(mo)l(dule)f(structur)l(e)d(of)k FB(p)p= Fr(th-p)l(ower)e (classes)i(of)697 4120 y(extensions)29 b(of)i(de)l(gr)l(e)l(e)f FB(p)p FE(.)d(T)-7 b(o)27 b(app)r(ear,)g(Israel)g(J.)g(Math.)456 4219 y([N])133 b(J.)35 b(Neukirc)n(h.)g Fr(F)-6= b(r)l(eie)38 b(Pr)l(o)l(dukte)f(pr)l(o-end)t(licher)j(Grupp)l(en)d(und)g(ihr)l(e)h (Kohomolo)l(gie)p FE(.)697 4319 y(Arc)n(h.)27 b(Math.)h(\(Basel\))f Fc(22)g FE(\(1971\),)f(337{357.)456 4419 y([Rib])65 b(P)-7 b(.)38 b(Rib)r(en)n(b)r(oim.)h= Fr(Some)h(examples)g(of)h(value)l(d)g (\014elds)p FE(.)e(J.)f(of)g(Algebra)g Fc(173)g FE(\(1995\),)697 4518 y(668{678.)456 4618 y([Ris])78 b(L.)34= b(Rib)r(es.)i Fr(Intr)l(o)l(duction)f(to)i(pr)l(o\014nite)f(gr)l(oups)h(and)g(Galois) h(c)l(ohomolo)l(gy)p FE(.)f(Queen's)697 4717= y(P)n(ap)r(ers)31 b(in)j(Pure)f(and)g(Applied)h(Mathematics)f Fc(24)p= FE(.)h(Kingston,)g (Canada:)47 b(Queen's)697 4817 y(Univ)n(ersit)n(y)-7 b(,)26 b(1970.)456 4917 y([S])149 b(J.-P)-7 b(.)31= b(Serre.)h Fr(L)l(o)l(c)l(al)i(Fields)p FE(.)g(T)-7 b(rans.)32= b(Marvin)f(Ja)n(y)g (Green)n(b)r(erg.)g(Graduate)h(T)-7 b(exts)32 b(in)697 5016 y(Mathematics)27 b Fc(67)p FE(.)g(New)h(Y)-7= b(ork-Berlin:)35 b(Springer-V)-7 b(erlag,)25 b(1979.)456 5116 y([W])110 b(A.)24 b(W)-7 b(adsw)n(orth.)23 b FB(p)p Fr(-Henselian)j(\014elds:)37 b FB(K)6 b Fr(-the)l(ory,)28= b(Galois)g(c)l(ohomolo)l(gy,)h(and)e(gr)l (ade)l(d)697 5216 y(Witt)i(rings.)f FE(P)n(aci\014c)f(J.)g(Math.)h Fc(105)f FE(\(1983\),)f(473{496.)p eop %%Page: 18 18 18 17 bop 456 255 a Fq(18)759 b(J)1340 236 y(\023)1330 255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y FC(Dep)j(ar)g(tment)51 b(of)e(Ma)-6 b(thema)g(tics,)55 b(Middlesex)49 b(College,)102 b(University)49 b(of)456 550 y(Western)31 b(Ont)-6 b(ario,)32 b(London,)d(Ont)-6 b(ario)63 b(N6A)31 b(5B7)62 b(CANAD)n(A)555 732 y= Fr(E-mail)31 b(addr)l(ess)7 b FE(:)38 b Fa(minac@uwo.ca)555 985 y FC(Dep)-6 b(ar)g(tment)27 b(of)e(Ma)-6 b(thema)g(tics,)27 b(D)n(a)-7 b(vidson)23 b(College,)j(Bo)n(x)e(7046,)h(D)n(a)-7 b(vidson,)456 1084 y(Nor)h(th)31 b(Car)n(olina)62 b(28035-7046)g(USA) 555 1267 y Fr(E-mail)31 b(addr)l(ess)7 b FE(:)38 b= Fa(joswallow@davids) o(on)o(.ed)o(u)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF --=====================_1049522553==_ 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