\documentclass[12pt,reqno]{amsart} \DeclareMathOperator{\ann}{ann}% \DeclareMathOperator{\chr}{char}% \DeclareMathOperator{\rank}{rank} \usepackage{amssymb} \usepackage[arrow,matrix]{xy} \newcommand{\F}{\mathbb{F}} \newcommand{\Fp}{\F_p} \newcommand{\Gal}{\text{\rm Gal}} \newcommand{\Ic}{\mathcal{I}} \newcommand{\im}{\text{\rm im}} \newcommand{\N}{\mathbb{N}} \newcommand{\sep}{\text{\rm sep}} \newcommand{\Z}{\mathbb{Z}} \begin{document} \title{Galois Module Structure of Galois Cohomology} \author[Nicole Lemire]{Nicole Lemire$^\star$} \address{Department of Mathematics, Middlesex College, \ University of Western Ontario, London, Ontario \ N6A 5B7 \ CANADA} \thanks{$^\star$Research supported in part by NSERC grant R3276A01.} \email{nlemire@uwo.ca} \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 NSERC grant R0370A01.} \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} Let $F$ be a field containing a primitive $p$th root of unity, and let $U$ be an open normal subgroup of index $p$ of the absolute Galois group $G_F$ of $F$. We determine the structure of the cohomology group $H^n(U,\Fp)$ as an $\Fp[G_F/U]$-module for all $n\in\mathbb{N}$. Previously this structure was known only for $n=1$, and until recently the structure even of $H^1(U,\Fp)$ was determined only for $F$ a local field, a case settled by Borevi\v{c} and Faddeev in the 1960s. \end{abstract} \date{October 29, 2004} \maketitle \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \theoremstyle{definition} \newtheorem*{remark*}{Remark} \parskip=10pt plus 2pt minus 2pt Let $F$ be a field containing a primitive $p$th root of unity $\xi_p$. Let $G_F$ be the absolute Galois group of $F$, $U$ an open normal subgroup of $G_F$ of index $p$, and $G=G_F/U$. Let $E$ be the fixed field of $U$ in the separable closure $F_\sep$ of $F$. Fix $a\in F$ such that $E=F(\root{p}\of{a})$, and let $\sigma\in G$ satisfy $\root{p}\of{a}^{\sigma-1}=\xi_p$. In the 1960s Z.~I.~Borevi\v{c} and D.~K.~Faddeev classified the possible $G$-module structures of the first cohomology groups $H^1(U, \Fp)$ in the case $F$ a local field \cite{B}. Quite recently this result was extended for all fields $F$ as above \cite{MS}. For the study of Galois cohomology it is important to extend these results to all cohomology groups $H^n(U, \Fp)$, $n\in\N$, and a solution of this problem was out of reach until now. Recently, based on earlier work of A.~S.~Merkurjev, M.~Rost and A.~A.~Suslin, V.~Voevodsky established the Bloch-Kato Conjecture \cite{Vo1,Vo2}, and it turns out that some of the main theorems in his proof are sufficient to determine the structure of all $G$-Galois modules $H^n(U, \Fp)$, using only simple arithmetical invariants attached to the field extension $E/F$. The theorems we use (quoted as Theorems~\ref{th:bk} and \ref{th:es} in section~\ref{se:bkktheory} below) had, in fact, been standard conjectures on Galois cohomology. It is interesting to point out, however, that the case $n=2$ could have already been settled some 20 years ago, thanks to the work of Merkurjev and Suslin \cite{MeSu}. The main ingredient for our determination of the $G$-module structure of $H^n(U,\Fp)$ is Milnor $K$-theory. (See \cite{Mi} and \cite[Chap.~IX]{FV}.) For $i\ge 0$, let $K_iF$ denote the $i$th Milnor $K$-group of the field $F$, with standard generators denoted by $\{f_1,\dots,f_i\}$, $f_1, \dots, f_i\in F\setminus \{0\}$. For $\alpha\in K_iF$, we denote by $\bar\alpha$ the class of $\alpha$ modulo $p$, and we use the usual abbreviation $k_nF$ for $K_nF/pK_nF$. We write $N_{E/F}$ for the norm map $K_nE\to K_nF$, and we use the same notation for the induced map modulo $p$. We denote by $i_E$ the natural homomorphism in the reverse direction. We also apply the same notation $N_{E/F}$ and $i_E$ for the corresponding homomorphisms between cohomology groups. The image of an element $\alpha\in K_i F$ in $H^i(G_F,\Fp)$ we also denote by $\alpha$. Voevodsky's proof of the Bloch-Kato Conjecture establishes a $G_F$-isomorphism $H^n(U,\Fp)\cong k_n E$. We formulate our results in terms of Galois cohomology for intended applications, but we use Milnor $K$-theory in our proof. We concentrate upon the case when $F$ is a perfect field, and at the end of the paper we indicate how one may reduce the case of an imperfect field $F$ to the case of characteristic $0$. Our decomposition depends on four arithmetic invariants $\Upsilon_1$, $\Upsilon_2$, $y$, $z$, which we define as follows. First, for an element $\bar\alpha$ of $k_i F$, let \begin{equation*} \ann_{k_{n-1}F}\bar\alpha = \ann \left(k_{n-1}F \xrightarrow{\bar\alpha \cdot -} k_{n-1+i}F\right) \end{equation*} denote the annihilator of the product with $\bar\alpha$. When the domain of $\bar\alpha$ is clear, we omit the subscript on the map and write simply $\ann\bar\alpha$. Because we will often use the elements $\overline{\{a\}}$, $\overline{\{\xi_p\}}$, $\overline{\{a,a\}}$, and $\overline{\{a, \xi_p\}}$, we omit the bars for these elements. We also omit the bar in the element $\overline{\{\root{p}\of{a}\}}\in k_n E$. Fix $n\in \N$ and $U$ an open normal subgroup of $G_F$ of index $p$ with fixed field $E$. Define invariants associated to $E/F$ and $n$ as follows: \begin{align*} d &:= \dim_{\Fp}\ k_n F\ /\ N_{E/F}k_n E \\ e &:= \dim_{\Fp}\ N_{E/F}k_n E \\ \Upsilon_1 &:= \dim_{\Fp}\ \ann \{a, \xi_p\}\ /\ \ann \{a\} \\ \Upsilon_2 &:= \dim_{\Fp}\ k_{n-1} F\ /\ \ann \{a,\xi_p\} \end{align*} \begin{equation*} y := \begin{cases} \dim_{\Fp}\ (N_{E/F}k_n E)\ /\ \{a\}\cdot k_{n-1}F, & p>2 \\ \dim_{\F_2}\ (N_{E/F}k_n E)\ /\ \{a\} \cdot \ann_{k_{n-1}F}\{a,-1\}, &p=2 \end{cases} \end{equation*} \begin{equation*} z := \begin{cases} \dim_{\Fp}\ (k_n F)\ /\ \left(\{\xi_p\}\cdot W + N_{E/F}k_n E\right), &p>2 \\ \dim_{\F_2}\ (k_n F)\ /\ \left(\{a\} \cdot k_{n-1}F + N_{E/F}k_n E\right), &p=2, \end{cases} \end{equation*} where $W$ is a complement of $\ann \{a, \xi_p\}$ in $k_{n-1}F$. (In Lemma~\ref{le:windep} we show that $z$ is independent of the choice of $W$.) Our main results are then the following. \begin{theorem}\label{th:pnot2} If $p>2$, $F$ is perfect and $n\in\N$ then \begin{equation*} H^n(U, \Fp) \cong X_1 \oplus X_2 \oplus Y \oplus Z \end{equation*} where \begin{enumerate} \item $X_1$ is a trivial $\Fp[G]$-module of dimension $\Upsilon_1$ \item $X_2$ is a direct sum of $\Upsilon_2$ cyclic $\Fp[G]$-modules of dimension 2 \item $Y$ is a free $\Fp[G]$-module of rank $y$ \item $Z$ is a trivial $\Fp[G]$-module of dimension $z$. \end{enumerate} Further we have \begin{enumerate} \setcounter{enumi}{4} \item $Y^G = i_E N_{E/F}H^n(U,\Fp)$ \item $N_{E/F}\colon X_1\oplus X_2 \to \{a\}\cdot H^{n-1}(G_F,\Fp)$ is surjective \item $\Upsilon_1+\Upsilon_2+y=e$ \item $\Upsilon_2+z=d$ \end{enumerate} \end{theorem} \begin{theorem}\label{th:p2} If $p=2$ and $n\in\N$ then \begin{equation*} H^n(U, \F_2) \cong X_1 \oplus Y \oplus Z \end{equation*} where \begin{enumerate} \item $X_1$ is a trivial $\F_2[G]$-module of dimension $\Upsilon_1$ \item $Y$ is a free $\F_2[G]$-module of rank $y$. \item $Z$ is a trivial $\F_2[G]$-module of dimension $z$. \end{enumerate} Further we have \begin{enumerate} \setcounter{enumi}{3} \item $Y^G = i_E N_{E/F}H^n(U,\F_2)$ \item $N_{E/F}\colon X_1 \to \{a\}\cdot \ann \{a, -1\}$ is an isomorphism \item $\Upsilon_1+y=e$ \item $\Upsilon_2+z=d$ \end{enumerate} \end{theorem} \begin{remark*} As explained at the end of section~\ref{se:esext}, if $n\leq 2$ in Theorem~\ref{th:pnot2}, we may remove the hypothesis that $F$ is perfect. The case $n=1$ in the two theorems recovers the results of \cite{MS}. \end{remark*} \section{Bloch-Kato and Milnor $K$-theory}\label{se:bkktheory} Our proof relies on the following two results in Voevodsky's proof of the Bloch-Kato Conjecture. Because we apply Voevodsky's results in the case when the base field contains a primitive $p$th root of unity we shall formulate Voevodsky's results restricted to this case. The first is the Bloch-Kato Conjecture itself: \begin{theorem}[{\cite[Def.~5.1]{Vo1} and \cite[Thm.~7.1]{Vo2}}] \label{th:bk}\ \begin{enumerate} \item Let $F$ be a field containing a primitive $p$th root of unity and $m\in\N$. Then the norm residue homomorphism \begin{equation*} k_m F\to H^m(G_F, \mu_p) \end{equation*} is an isomorphism. \item For any cyclic extension $E/F$ of degree $p$, the sequence \begin{equation*} K_mE \xrightarrow{\sigma-1} K_mE \xrightarrow{N_{E/F}} K_mF \end{equation*} is exact. \end{enumerate} \end{theorem} \noindent The second result establishes an exact sequence connecting $k_mF$ and $k_mE$ for consecutive $m$. (We translate the statement of the original result to $K$-theory using the previous theorem.) In the following result $a$ is chosen to satisfy $E=F(\root{p}\of{a})$. \begin{theorem}[{\cite[Prop.~5.2]{Vo1}}]\label{th:es} Let $F$ be a field containing a primitive $p$th root of unity with no extensions of degree prime to $p$. Then for any cyclic extension $E/F$ of degree $p$ and $m\ge 1$, the sequence \begin{equation*} k_{m-1}E \xrightarrow{N_{E/F}} k_{m-1}F \xrightarrow{\{a\} \cdot -} k_m F \xrightarrow{i_E} k_m E \end{equation*} is exact. \end{theorem} Now if $F$ is a perfect field, we observe that we may remove the hypothesis that the field $F$ has no extensions of degree prime to $p$. It is precisely to ensure that the sequence above is exact that we require the hypothesis that $F$ is perfect in Theorem~\ref{th:pnot2}. We give a proof of the following theorem in section~\ref{se:esext}. \begin{theorem}[{Modification of Theorem~\ref{th:es}}]\label{th:esext} Let $F$ be a field containing a primitive $p$th root of unity, and if $p>2$ assume that $F$ is perfect. Then for any cyclic extension $E/F$ of degree $p$ and $m\geq 1$ the sequence \begin{equation*} k_{m-1}E \xrightarrow{N_{E/F}} k_{m-1}F \xrightarrow{\{a\} \cdot -} k_m F \xrightarrow{i_E} k_m E \end{equation*} is exact. \end{theorem} \section{Notation and Lemmas} For the remainder of the paper, except for section~\ref{Red}, we assume that $F$ is perfect if $p>2$. We fix $n\in \N$ and the cyclic extension $E=F(\root{p}\of{a})$, and we write $k_{n-1}F = \ann\{a\} \oplus V \oplus W$, where $\ann\{a, \xi_p\} = \ann\{a\} \oplus V$. Observe that $\Upsilon_1=\dim_{\Fp} V$ and $\Upsilon_2=\dim_{\Fp} W$. We show that the invariant $z$ is independent of the choice of $W$: \begin{lemma}\label{le:windep} The invariant $z$ is independent of the particular complement $W$ of $\ann \{a, \xi_p\}$ in $k_{n-1}F$. \end{lemma} \begin{proof} This is obvious for $p=2$, so assume that $p>2$. Let $W'$ be another $\Fp$-subspace of $k_{n-1}F$ with $k_{n-1}F = \ann\{a, \xi_p\}\oplus W'$. Then we may choose bases $\{w_i\}_{i\in \Ic}$ and $\{w'_i\}_{i\in \Ic}$ of $W$ and $W'$, respectively, such that $w'_i=w_i+u_i$ for $u_i \in \ann \{a,\xi_p\}$. Then $\{\xi_p\}\cdot w'_i = \{\xi_p\}\cdot w_i + \{\xi_p\}\cdot u_i$, and $\{\xi_p\}\cdot u_i \in \ann\{a\}$. Because $\ann\{a\} = N_{E/F}k_n E$ by Theorem~\ref{th:esext}, we see that \begin{equation*} \{\xi_p\}\cdot W + N_{E/F}k_nE = \{\xi_p\}\cdot W' + N_{E/F}k_nE \end{equation*} and therefore $z$ does not depend on the choice of $W$. \end{proof} We denote by $i_E\colon K_nF\to K_nE$ the map induced by the inclusion of $F$ in $E$. In what follows we will frequently refer to the element $\root{p}\of{a}$, and so we abbreviate it by $A$. We will also often use the observation that if $p=2$ then $\{a,\xi_p\}=\{a,-1\}=\{a,a\}\in k_2F$, while if $p>2$ then $\{a,a\} = 0\in k_2 F$. Finally, we will use the projection formula for taking the norms of standard generators of $K_i F$ (see \cite[p.~81]{FW}). \begin{lemma}\label{le:VW} We have the vector space isomorphism \begin{equation*} V\oplus W \xrightarrow{\{a\}\cdot -} \{a\}\cdot k_{n-1}F \end{equation*} and, if $p>2$, the compositum of the maps $\{\xi_p\}\cdot-$ and $i_E$ \begin{equation*} W \xrightarrow{\{\xi_p\}\cdot - } \{\xi_p\} \cdot W \xrightarrow{i_E} i_E(\{\xi_p\}\cdot W) \end{equation*} is a vector space isomorphism as well. \end{lemma} \begin{proof} The first isomorphism follows from the fact that $V\oplus W$ is a complement in $k_{n-1}F$ of the kernel of multiplication by $\{a\}$. For the second, assume $p>2$. Suppose that $\bar w\in W$ and $\bar\alpha = \{\xi_p\}\cdot \bar w\in \ker i_E$. Then by Theorem~\ref{th:esext}, $\bar\alpha = \{a\}\cdot \bar c$ for $c\in K_{n-1}F$. Since $\{a,a\}=0$ we see that $\{a\}\cdot \bar\alpha = 0$. But then $\bar w \in \ann \{a,\xi_p\}$ and so $\bar w=0$. \end{proof} For $\gamma\in K_nE$, let $l(\gamma)$ denote the dimension of the cyclic $\Fp[G]$-submodule $\langle \bar \gamma \rangle$ of $k_n E$ generated by $\bar\gamma$. Then we have \begin{equation*} (\sigma-1)^{l(\gamma)-1}\langle \bar\gamma \rangle = \langle \bar\gamma\rangle^G \neq 0 \text{\ \ \ and \ \ } (\sigma-1)^{l(\gamma)}\langle \bar\gamma \rangle=0. \end{equation*} We denote by $N$ the map $(\sigma-1)^{p-1}$ on $k_n E$. Because $(\sigma-1)^{p-1}=1+\sigma+\dots+\sigma^{p-1}$ in $\Fp[G]$, we may use $i_EN_{E/F}$ and $N$ interchangeably on $k_n E$. \begin{lemma}\label{le:extend} Suppose $p>2$ and $\gamma\in K_n E$. \begin{enumerate} \item If $3\le l(\gamma) \le p$, then there exists $\alpha \in K_nE$ such that \begin{equation*} \langle N\bar\alpha \rangle = \langle \bar \gamma \rangle^G. \end{equation*} \item If $l(\gamma)=2$ and \begin{equation*} \bar \gamma \not\in \{A\} \cdot i_E(k_{n-1}F) + (k_n E)^G \end{equation*} then there exist $\alpha\in K_nE$ and $b\in K_{n-1}F$ such that \begin{equation*} \langle N\bar\alpha \rangle = \langle \bar \gamma + \{A\}\cdot i_E(\bar b) \rangle^G. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} Let $l=l(\gamma)$ and suppose $3\le l\le i\le p$. We show by induction on $i$ that there exists $\alpha_i\in K_nE$ such that $\left<(\sigma-1)^{i-1} \bar\alpha_i\right> = \langle \bar\gamma \rangle^G$. Then setting $\alpha:=\alpha_p$, the proof will be complete. If $i=l$ then $\alpha_i=\gamma$ suffices. Assume now that $l\le i = \langle \bar\gamma \rangle^G$ and we can set $\alpha_{i+1}=\omega$. For the second part, suppose $l=2=i$. Proceeding in the same way as above, we see that for $\alpha_2=\gamma$ we have $N_{E/F}\alpha_2 = \{a\}\cdot b + pf $ for $b\in K_{n-1}F$ and $f\in K_nF$. As before, there exists $\omega\in K_nE$ such that $(\sigma-1)\omega = \alpha_2 - (\{A\}\cdot i_E(b) + i_E(f))$. Then \begin{equation*} (\sigma-1)^2\omega = (\sigma-1)(\alpha_2-\{A\} \cdot i_E(b)) = (\sigma-1)(\gamma-\{A\} \cdot i_E(b)). \end{equation*} Observe that $\bar\gamma-\{A\}\cdot i_E(\bar b)\not\in (k_n E)^G$ by hypothesis. Therefore $l(\gamma-\{A\}\cdot i_E(b))=2$ and we can set $\alpha_3:=\omega$. We may then continue by induction on $i$ as above, concluding that there exists an element $\alpha = \alpha_p\in K_nE$ such that $\left = \left<(\sigma-1)^{p-1} \bar\alpha_p\right> = \langle \bar\gamma - \{A\}\cdot i_E(\bar b)\rangle^G$, as required. \end{proof} In the following lemma we elongate the exact sequence of Theorem~\ref{th:esext}. \begin{lemma}\label{le:fixedes} The following sequence is exact: \begin{equation*} 0\to\ann\{a\}\to k_{n-1}F \xrightarrow{\{a\}\cdot -} k_n F \xrightarrow{i_E} (k_n E)^G \xrightarrow{N_{E/F}} \{a\}\cdot \ann \{a,\xi_p\}\to 0. \end{equation*} Here the map $\ann\{a\}\to k_{n-1}F$ is the natural inclusion. \end{lemma} \begin{proof} We show first that $N_{E/F}((k_n E)^G)\subset \{a\} \cdot \ann \{a,\xi_p\}$. Let $\bar \alpha\in (k_n E)^G$ and $\beta = N_{E/F}\alpha$. Since $i_E(N_{E/F}\bar\alpha) = (\sigma-1)^{p-1} \bar\alpha= 0$ we have that $\bar\beta = N_{E/F}\bar \alpha = \{a\}\cdot \bar b$ for some $b\in K_{n-1}F$ by Theorem~\ref{th:esext}. Suppose $p=2$. Since $\bar\beta$ is in the image of $N_{E/F}$, we have by Theorem~\ref{th:esext} that $\{a\} \cdot \bar \beta = \{a,a\} \cdot \bar b = 0$. Since $\{a,a\} = \{a,-1\}$, we have $\bar b \in \ann \{a,-1\}$. Now suppose that $p>2$. Write $\beta = \{a\} \cdot b + pf$ for some $f\in K_nF$. Then by the projection formula \begin{equation*} N_{E/F}\big( \alpha - (\{A\} \cdot i_E(b) + i_E(f)) \big) = 0. \end{equation*} By Theorem~\ref{th:bk}, there exists $\omega\in K_nE$ such that \begin{equation*} (\sigma-1)\omega = \alpha - (\{A\} \cdot i_E(b) - i_E(f)). \end{equation*} Then $(\sigma-1)^2\bar\omega = \{\xi_p\} \cdot i_E(\bar{b})$. If $(\sigma-1)^2\bar\omega = 0$ then since by Theorem~\ref{th:esext}, $\ker i_E = \{a\} \cdot k_{n-1}F$, \begin{equation*} \{\xi_p\} \cdot \bar b = \{a\} \cdot \bar h \end{equation*} for some $h\in K_{n-1}F$. Because $\{a,a\}=0$, the right-hand side of the preceding equation is annihilated by $\{a\}$. Therefore $\bar b\in \ann \{a,\xi_p\}$. If $(\sigma-1)^2\bar \omega \neq 0$ then $l(\omega)=3$ and Lemma~\ref{le:extend} shows that \begin{equation*} i_E(\{\xi_p\}\cdot \bar b) = cN\bar\lambda = i_E(N_{E/F}(\overline{c\lambda})) \end{equation*} for some $\lambda\in K_nE$ and $c\in \Z$. Since by Theorem~\ref{th:esext}, $\ker i_E = \{a\}\cdot k_{n-1}F$ we have \begin{equation*} \{\xi_p\} \cdot \bar b = N_{E/F}(\overline{c\lambda}) + \{a\} \cdot \bar h \end{equation*} for some $h\in K_{n-1}F$. Now by Theorem~\ref{th:esext} and the fact that $\{a,a\}=0$, the right-hand side of the preceding equation is annihilated by $\{a\}$. Then $\bar b\in \ann \{a,\xi_p\}$. Hence in all cases $N_{E/F}\bar\alpha \in \{a\}\cdot \ann \{a, \xi_p\}$. Exactness at the first two terms is obvious, and exactness at the third term follows from Theorem~\ref{th:esext}. For exactness at the fourth term, suppose $$\bar \gamma\in (k_n E)^G\mbox{ and }N_{E/F}\bar\gamma = 0.$$ Then $N_{E/F}\gamma= pf$ for $f\in K_nF$. Let $\beta=\gamma-i_E(f)$. Then $N_{E/F}\beta=0$ and by Theorem~\ref{th:bk} there exists $\alpha\in K_nE$ such that $(\sigma-1)\alpha = \beta$. If $p=2$ then $\bar\beta=i_E(N_{E/F}\bar\alpha)\in i_E k_n F$ and we are done. Thus assume $p>2$. Now suppose $\bar\alpha \in \{A\}\cdot i_E(k_{n-1}F) + (k_n E)^G$. Then \begin{equation*} \bar\beta = (\sigma-1)\bar \alpha \in \{\xi_p\} \cdot i_E(k_{n-1} F) \subset i_E(k_n F), \end{equation*} and hence $\bar\gamma = \bar\beta + i_E(\bar f)\in i_E(k_n F)$ as well. Otherwise $\bar\alpha\notin\{A\}\cdot i_E(k_{n-1}F)+(k_n E)^G$. Now if $(\sigma-1)\bar\alpha=\bar\beta=0$ we are done as then $\bar\gamma=i_E(\bar f)$. Hence assume $(\sigma-1)\bar\alpha\neq 0$. Then $l(\alpha)=2$ and by Lemma~\ref{le:extend} we see that there exist $\delta\in K_n E,b\in K_{n-1}F$ and $c\in\Z$ such that \begin{equation*} c N \bar\delta = (\sigma-1)(\bar\alpha+\{A\}\cdot i_E(\bar{b}))=\bar\beta+\{\xi_p\}\cdot i_E(\bar{b}). \end{equation*} Thus $\bar\beta = c N \bar\delta-\{\xi_p\}\cdot i_E(\bar{b})\in i_E(k_n F)$ and exactness at the fourth term is established. Finally we show the exactness at the fifth term. Since \begin{equation*} \{a\}\cdot\ann\{a,\xi_p\}=\{a\}\cdot V \end{equation*} \noindent it is enough to show that each element $\{a\}\cdot\bar{v}$ where $\bar{v}\in V$ can be written as $N_{E/F}\bar\alpha$ for some $\bar\alpha\in(k_n E)^G$. Observe that $(\sigma-1)(\{A\}\cdot i_E\bar{v})=\{\xi_p\}\cdot i_E(\bar{v})$. Also we have \begin{equation*} N_{E/F}(\{A\}\cdot i_E (\bar v))= \begin{cases} \{a\}\cdot\bar v &\mbox{ if }p>2 \\ \{-a\}\cdot\bar v &\mbox{ if }p=2. \end{cases} \end{equation*} Therefore it is enough to show that there exists an element $\bar\gamma\in k_n E$ such that $(\sigma-1)\bar\gamma=\{\xi_p\}\cdot i_E(\bar{v})$ and \begin{equation*} N_{E/F}\bar\gamma = \begin{cases} 0 &\mbox{ if }p>2 \\ \{-1\}\cdot\bar v &\mbox{ if }p=2. \end{cases} \end{equation*} Indeed then we can set $\bar\alpha=\{A\}\cdot i_E(\bar{v})-\bar\gamma$. Because $\bar{v}\in\ann\{a,\xi_p\}$ we see that $\{\xi_p\}\cdot i_E(\bar{v})\in\ann\{a\}$. By Theorem~\ref{th:esext} there exists $\bar\beta\in k_n E$ such that \begin{equation*} \{\xi_p\}\cdot\bar{v}=N_{E/F}\bar\beta\mbox{ and } i_E(N_{E/F}\bar\beta)=(\sigma-1)^{p-1}\bar\beta. \end{equation*} Then setting $\bar\gamma=(\sigma-1)^{p-2}\bar\beta$ we obtain our required element. The proof of our lemma has now been completed. \end{proof} Finally, we need a general lemma about $\Fp[G]$-modules. \begin{lemma}[Exclusion Lemma]\label{le:excl} Let $M_1$ and $M_2$ be $\Fp[G]$-modules contained in a common $\Fp[G]$-module. Suppose that $M_1^G\cap M_2^G=\{0\}$. Then $M_1+M_2=M_1\oplus M_2$. \end{lemma} \begin{proof} Let $M=M_1 \cap M_2$ and suppose that $m\in M\setminus \{0\}$. Let \begin{equation*} \tilde m=(\sigma-1)^{l(m)-1}(m)\neq 0. \end{equation*} Then $\tilde m\in M_1^G\cap M_2^G$, a contradiction. Hence $M_1\cap M_2=\{0\}$ and $M_1+M_2= M_1\oplus M_2$. \end{proof} \section{Construction of Submodules} \begin{proposition}\label{pr:x} $k_n E$ contains a submodule $X_1$ such that \begin{itemize} \item $X_1$ is a trivial $\Fp[G]$-module of dimension $\Upsilon_1$ \item $X_1 \cap i_E k_n F = \{0\}$ \item $N_{E/F}$ restricts to an isomorphism $X_1\to \{a\}\cdot V$. \end{itemize} Moreover, if $p>2$, then $k_n E$ contains a submodule $X_2$, independent of $X_1$, such that \begin{itemize} \item $X_2$ is a direct sum of $\Upsilon_2$ cyclic submodules of dimension $2$ and $\dim_{\Fp}X_2^G=\Upsilon_2$. \item $(X_1+X_2) \cap i_E k_n F = (\sigma-1)X_2 = X_2^G = i_E(\{\xi_p\}\cdot W)$ \item We have an exact sequence \begin{equation*} 0\to \{\xi_p\} \cdot W \xrightarrow{i_E} X_1+X_2 \xrightarrow{N_{E/F}} \{a\} \cdot k_{n-1}F \to 0 \end{equation*} \end{itemize} \end{proposition} \begin{proof} Let $\Ic$ be an $\Fp$-basis for $V$. Let $\bar v$ be an arbitrary element of $\Ic$, and consider $\bar\alpha = \{A\} \cdot i_E\bar v$. Now $(\sigma-1)\bar\alpha = i_E(\{\xi_p\}\cdot \bar v)$. Since $\bar v\in \ann \{a,\xi_p\}$ we see that $\{\xi_p\}\cdot\bar v\in\ann_{k_n F}\{a\}$. By Theorem~\ref{th:esext} \begin{equation*} \{\xi_p\}\cdot \bar v = N_{E/F}\bar\beta\mbox{ and }i_E(N_{E/F}\bar\beta)= N\bar\beta=(\sigma-1)^{p-1}\bar\beta \end{equation*} for some $\beta\in K_n E$. Set $\gamma = (\sigma-1)^{p-2}\beta$ and $\bar x_v= \bar\alpha - \bar\gamma \in k_n E$. If $p=2$ then \begin{equation*} N_{E/F}\bar x_v =\{-a\} \cdot \bar v - N_{E/F}\bar\gamma = \{-a\} \cdot \bar v - \{-1\}\cdot \bar v = \{a\}\cdot \bar v. \end{equation*} If $p>2$, then observe that since $\gamma$ is in the image of $\sigma-1$ we have $\overline{N_{E/F}\gamma}=0$. Then, by the projection formula \begin{equation*} N_{E/F}\bar x_v = \{a\}\cdot \bar v - \overline{N_{E/F}\gamma} = \{a\} \cdot \bar v. \end{equation*} Now in either case, since $(\sigma-1)^{p-1}\bar\beta = i_E(N_{E/F}\bar\beta)$, \begin{equation*} (\sigma-1)\bar x_v = i_E(\{\xi_p\}\cdot \bar v) - (\sigma-1)^{p-1} \bar\beta \\ = i_E(\{\xi_p\}\cdot \bar v) - i_E(\{\xi_p\}\cdot \bar v) = 0. \end{equation*} Set \begin{equation*} X_1 := \oplus_{\bar v\in \Ic} \langle \bar x_v\rangle. \end{equation*} We have shown that $X_1$ is a trivial $\Fp[G]$-module. Moreover, because $N_{E/F}\bar x_v=\{a\}\cdot \bar v$ and $\{a\}\cdot -$ is injective on $V$ by Lemma~\ref{le:VW}, \begin{equation*} N_{E/F}\big\vert_{X_1}\colon X_1\to \{a\}\cdot V \end{equation*} takes a basis of $X_1$ to a basis of $\{a\}\cdot V$ and $\dim_{\Fp} X_1 = \dim_{\Fp} V = \Upsilon_1$. Finally, since $N_{E/F}$ is trivial on $i_E k_n F$, we have $X_1\cap i_E k_n F = \{0\}$. Now suppose that $p>2$. Set \begin{equation*} X_2 := \left(\{A\}\cdot i_EW\right) + i_E(\{\xi_p\} \cdot W). \end{equation*} Let $\bar w\in W$ and consider $\bar x_w=\{A\} \cdot i_E(\bar w)$. Since $(\sigma-1)\bar x_w = i_E(\{\xi_p\}\cdot \bar w)$ and $(\sigma-1)(\{\xi_p\} \cdot \bar w) = 0$, we obtain $(\sigma-1)X_2 = i_E(\{\xi_p\}\cdot W)$. Hence on $\{A\}\cdot i_E W$, $\sigma-1$ acts as $i_E(\{\xi_p\}\cdot -)$, which by Lemma~\ref{le:VW} is an isomorphism of vector spaces. Hence $\sigma-1$ is an isomorphism as well. Moreover, if an arbitrary $\{A\}\cdot i_E (\bar w_1) + \{\xi_p\} \cdot i_E (\bar w_2)\in X_2$ lies in the kernel of $\sigma-1$, $\bar w_1=0$. Hence $X_2^G = i_E(\{\xi_p\}\cdot W)$. Since we already observed that $X_1\cap i_E k_n F=\{0\}$ we see that $X_2^G\cap X_1=\{0\}$ and by Lemma~\ref{le:excl} we conclude that $X_1+X_2=X_1\oplus X_2$. By the projection formula $N_{E/F}\bar x_w = \{a\}\cdot \bar w$ and by the definition of $W$, $\{a\}\cdot \bar w = 0$ implies $\bar w = 0$. Since $N_{E/F}(\{\xi_p\}\cdot i_E (\bar w_2)) = 0$ for all $\bar w_2\in W$, we deduce that restricted to $X_2$, $N_{E/F}$ surjects $X_2$ onto $\{a\}\cdot W$ with kernel $i_E(\{\xi_p\}\cdot W)$. By Lemma~\ref{le:VW}, $\{a\}\cdot k_{n-1}F = \{a\}\cdot (V+W)$; hence on $X_1\oplus X_2$, $N_{E/F}$ is a surjection onto $\{a\}\cdot k_{n-1}F$ with kernel $i_E(\{\xi_p\} \cdot W)$. Finally observe that $N_{E/F}i_E k_n F = \{0\}$. Hence \begin{equation*} (X_1+X_2) \cap i_E k_nF \subset i_E(\{\xi_p\}\cdot W). \end{equation*} Since $i_E(\{\xi_p\}\cdot W) \subset i_E k_n F$, we have equality. Now we have shown that $\sigma-1$ is an isomorphism of vector spaces $\{A\}\cdot i_E W \to i_E(\{\xi_p\}\cdot W)$, and by Lemma~\ref{le:VW}, we have an isomorphism $W\to i_E(\{\xi_p\}\cdot W)$. Therefore $X_2$ is a direct sum of cyclic submodules $\langle \bar x_w \rangle$ of dimension $2$, with $\bar x_w$ in one-to-one correspondence with basis elements of $W$. Hence the direct sum contains $\Upsilon_2$ cyclic summands. \end{proof} If $p=2$, let $X=X_1$ be a submodule of $k_n E$ satisfying the conditions of the preceding proposition. If $p>2$, let $X=X_1+X_2$ for $X_1$, $X_2$ satisfying the conditions of the same. \begin{proposition}\label{pr:y} $k_n E$ contains a submodule $Y$ independent from $X$ such that \begin{itemize} \item $Y$ is a free $\Fp[G]$-module of rank \begin{equation*} y = \begin{cases} \dim_{\Fp} (N_{E/F}k_n E)/\{a\}\cdot k_{n-1}F, &p>2 \\ \dim_{\F_2} (N_{E/F}k_n E)/\{a\}\cdot \ann_{k_{n-1}F} \{a,-1\}, &p=2 \end{cases} \end{equation*} \item $Y^G = i_E N_{E/F}k_nE$ \item if $p>2$, $\Upsilon_1 + \Upsilon_2 + y = e$ \item if $p=2$, $\Upsilon_1 + y = e$ \end{itemize} \end{proposition} \begin{proof} Let $\Ic$ be a basis for the subspace $i_E(N_{E/F}k_n E)$. For each basis element $\bar y\in \Ic$, let $\alpha_y\in K_nE$ satisfy $i_E(N_{E/F}\bar \alpha_y) = \bar y$. Then $\langle \bar\alpha_y \rangle$ is a cyclic submodule of dimension $p$, hence isomorphic to $\Fp[G]$, with \begin{equation*} \langle \bar\alpha_y\rangle^G = (\sigma-1)^{p-1}\langle \bar\alpha_y\rangle = \langle N\bar\alpha_y\rangle = \langle \bar y\rangle. \end{equation*} Set \begin{equation*} Y = \sum_{\bar y\in \Ic} \langle \bar\alpha_y \rangle. \end{equation*} By Lemma~\ref{le:excl}, $Y=\oplus_{\bar y\in \Ic} \langle \bar \alpha_y \rangle$ and so $Y$ is a free $\Fp[G]$-module. Moreover, \begin{equation*} Y^G=(\sigma-1)^{p-1}Y=NY=\oplus_{\bar y\in \Ic} \langle \bar y\rangle = i_E(N_{E/F}k_n E). \end{equation*} Now the rank of $Y$ is equal to the dimension of $i_E(N_{E/F}k_n E)$, or \begin{equation*} \dim_{\Fp} (N_{E/F}k_n E) / \left((N_{E/F}k_n E) \cap \ker i_E\right). \end{equation*} Now by Theorem~\ref{th:esext}, $N_{E/F}(k_n E) = \ann\{a\}$, and by the same Theorem, $\ker i_E = \{a\} \cdot k_{n-1}F$. Hence \begin{equation*} N_{E/F}(k_n E) \cap \ker i_E = \ann_{k_n F}\{a\} \cap \{a\}\cdot k_{n-1}F. \end{equation*} Suppose that $p=2$. Since $\{a,a\} = \{a,-1\}$ we deduce that \begin{equation*} N_{E/F}(k_n E) \cap \ker i_E = \{a\} \cdot \ann \{a,-1\}. \end{equation*} The dimension of this subspace is equal to $\dim_{\Fp} \ann \{a,-1\}/\ann \{a\}$, or $\Upsilon_1$. Now suppose that $p>2$. Since $\{a,a\}=0$, $\{a\}\cdot k_{n-1}F \subset \ann \{a\}$ and we deduce that \begin{equation*} N_{E/F}(k_n E) \cap \ker i_E = \{a\}\cdot k_{n-1}F, \end{equation*} which is of dimension $\dim_{\Fp} (k_{n-1}F)/\ann \{a\}$, or $\Upsilon_1+\Upsilon_2$. As $e=\dim_{\Fp} N_{E/F}k_n E$, we deduce that if $p=2$ then $\Upsilon_1 + \rank Y = e$ and if $p>2$, $\Upsilon_1+\Upsilon_2+\rank Y = e$. Now we claim that $Y$ is independent from $X$. Suppose $\bar \beta\in X^G\cap Y^G$. Now $Y^G = i_E(N_{E/F} k_n E) \subset i_E(k_n F)$, so $\bar\beta = i_E(\bar\alpha)$ where $\bar \alpha = N_{E/F}\bar\gamma$ for $\gamma\in K_n E$. If $p=2$ then by Proposition~\ref{pr:x}, $i_E(k_n F) \cap X = \{0\}$, and so $X\cap Y = \{0\}$ by Lemma~\ref{le:excl}. If $p>2$, Proposition~\ref{pr:x} tells us that \begin{equation*} i_E(k_n F) \cap X = X_2^G = \{\xi_p\} \cdot i_E W. \end{equation*} Hence $\bar \beta = \{\xi_p\} \cdot i_E(\bar w)$ for $\bar w\in W$. Since $i_E(\{\xi_p\} \cdot \bar w)=i_E(\bar\alpha)$ and by Theorem~\ref{th:esext}, $\ker i_E = \{a\} \cdot k_{n-1}F$, \begin{equation}\label{eq:y} \{\xi_p\} \cdot \bar w = \bar \alpha + (\{a\} \cdot \bar f) \end{equation} for $f\in K_{n-1}F$. Now because $\bar\alpha\in N_{E/F} k_n E$, by Theorem~\ref{th:esext}, $\{a\}\cdot \bar\alpha=0$. Moreover, $\{a,a\} = 0$ since we have assumed that $p>2$. Hence the right-hand side of \eqref{eq:y} is annihilated by multiplication by $\{a\}$. Therefore $\bar w\in \ann \{a,\xi_p\}$, and by the definition of $W$, $\bar w= 0$. By Lemma~\ref{le:excl}, $X+Y=X\oplus Y$. \end{proof} Now let $X$ and $Y$ be submodules satisfying the conditions of the preceding propositions. \begin{proposition}\label{pr:z} $k_n E$ contains a submodule $Z$ independent from $X+Y$ such that \begin{itemize} \item $Z$ is a trivial $\Fp[G]$-module of dimension \begin{equation*} z = \begin{cases} \dim_{\Fp}(k_n F)/(\{\xi_p\}\cdot W + N_{E/F}k_n E), &p>2 \\ \dim_{\F_2} (k_n F)/(\{a\}\cdot k_{n-1}F + N_{E/F}k_n E), &p=2 \end{cases} \end{equation*} \item $(k_n E)^G=X^G+Y^G+Z$ \item $\Upsilon_2 + z = d$ \end{itemize} \end{proposition} \begin{proof} Let $Z$ be a complement of $(X^G+Y^G)\cap i_E(k_n F)$ in $i_E(k_n F)$. By Lemma~\ref{le:excl}, $(X+Y)+Z = (X+Y)\oplus Z$. Clearly $X^G+Y^G+Z\subset (k_n E)^G$. Now suppose $\bar \alpha\in (k_n E)^G$ and let $\beta = N_{E/F}\alpha$. By Lemma~\ref{le:fixedes}, $\bar\beta = \{a\} \cdot \bar b$ for some $\bar b\in \ann \{a,\xi_p\}$. Let $\bar v\in V$ be the component of $\bar b$ in the decomposition $\ann \{a\} \oplus V$ of $\ann \{a,\xi_p\}$. By Proposition~\ref{pr:x}, there exists $\bar \gamma\in X_1\subset X^G$ such that \begin{equation*} N_{E/F}\bar \gamma = \{a\} \cdot \bar v = \{a\} \cdot \bar b = \bar \beta. \end{equation*} Then $N_{E/F}(\bar \alpha - \bar\gamma)=0$. By Lemma~\ref{le:fixedes}, $\bar\alpha-\bar\gamma\in i_E(k_n F)$. But $i_E(k_n F)\subset X^G+Y^G+Z$. Hence $\bar \alpha\in X^G+Y^G+Z$ and we have shown that $(k_n E)^G = X^G+Y^G+Z$. For the dimension of $Z$, assume first that $p>2$. By Theorem~\ref{th:esext}, $N_{E/F}k_n E = \ann_{k_n F} \{a\}$ and $\ker i_E = \{a\} \cdot k_{n-1}F$. Since $\{a,a\} = 0$ we see that $\ker i_E \subset N_{E/F}k_n E$. Hence \begin{equation*} d = \dim_{\Fp} \frac{k_n F}{N_{E/F}k_n E} = \dim_{\Fp} \frac{i_E(k_n F)}{i_E(N_{E/F}k_n E)}= \dim_{\Fp} \frac{i_E(k_n F)}{Y^G}, \end{equation*} where in the last equation we use Proposition~\ref{pr:y} to identify $Y^G$. By Propositions~\ref{pr:x} and \ref{pr:y}, $(X^G+Y^G)\cap i_E(k_n F) = X_2^G\oplus Y^G$. Hence $d=\dim_{\Fp} (X_2^G \oplus Y^G \oplus Z)/Y^G$. By Proposition~\ref{pr:x}, $\dim_{\Fp} X_2^G = \Upsilon_2$. Hence $\Upsilon_2 + \dim_{\Fp} Z = d$ for $p>2$. Also we see that \begin{equation*} \dim_{\Fp}Z = \dim_{\Fp}\frac{i_E (k_n F)}{X_2^G\oplus Y^G} = \dim_{\Fp} \frac{k_n F}{\{\xi_p\}\cdot W+N_{E/F}k_n E} = z. \end{equation*} Now assume $p=2$. By Propositions~\ref{pr:x} and \ref{pr:y}, $i_E(k_n F)\cap (X^G+Y^G) = Y^G$. Proceeding as in the last case, \begin{align*} \dim_{\F_2} Z &= \dim_{\F_2} \frac{i_E(k_n F)}{Y^G} = \dim_{\F_2} \frac{i_E(k_nF)}{i_E(N_{E/F}k_nE)} \\ &= \dim_{\F_2} \frac{k_nF}{N_{E/F}k_nE + \ker i_E} \\ &= \dim_{\F_2} \frac{k_n F}{N_{E/F}k_n E + \{a\}\cdot k_{n-1}F} = z, \end{align*} since $\ker i_E = \{a\}\cdot k_{n-1}F$, by Theorem~\ref{th:esext}. We then consider the filtration \begin{equation*} k_n F \supset \Big((\{a\}\cdot k_{n-1}F) + N_{E/F}k_n E\Big) \supset N_{E/F}k_n E. \end{equation*} The dimension of the quotient of the first and third modules is, by definition, $d$. By Theorem~\ref{th:esext}, $N_{E/F}k_n E = \ann_{k_n F} \{a\}$. Since $\{a,a\}=\{a,-1\}$ we see that \begin{equation*} (\{a\}\cdot k_{n-1}F) \cap N_{E/F}k_n E = \{a\} \cdot V. \end{equation*} Hence \begin{equation*} \dim_{\F_2} \frac{(\{a\}\cdot k_{n-1}F) + N_{E/F}k_n E}{ N_{E/F}k_n E} = \dim_{\F_2} \frac{\{a\} \cdot k_{n-1}F}{\{a\} \cdot V} = \dim_{\F_2} \{a\}\cdot W. \end{equation*} By Lemma~\ref{le:VW}, $\dim_{\F_2} \{a\}\cdot W = \dim_{\F_2} W = \Upsilon_2$. Hence $\Upsilon_2 + \dim_{\F_2} Z = d$ for $p=2$ as well. \end{proof} \section{Proofs of Theorems \ref{th:pnot2} and \ref{th:p2}} \begin{proof}[Proof of Theorem~\ref{th:pnot2}] By Propositions~\ref{pr:x}, \ref{pr:y}, and \ref{pr:z}, there exist independent submodules $X=X_1+X_2$, $Y$, and $Z$ satisfying the conditions of the theorem. All that remains is to show that $k_n E = X+Y+Z$. We proceed by induction on the length $l(\gamma)$ of the cyclic submodule $\langle \bar \gamma \rangle$ of $k_n E$ generated by an arbitrary element $\bar\gamma\in k_n E$. If $l(\gamma)=1$, then by Proposition~\ref{pr:z}, $\bar\gamma\in X^G+Y^G+Z$. Assume then that $\bar \beta\in X+Y+Z$ if $l(\beta)\le i2$ and $F$ is perfect. Let $S$ be any $p$-Sylow subgroup of $G_F=\Gal(F_{\sep}/F)$, and set $L$ to be the fixed field of $S$. Because $F$ is perfect, the separable closure $F_{\sep}$ is identical to the algebraic closure $\bar F$, and hence each finite extension of $L$ has degree a power of $p$. In particular, all of the hypotheses of Theorem~\ref{th:es} are valid for the field $L$ in place of $F$. Furthermore, $([L:F],p)=1$. (Here we use basic properties of supernatural numbers and Sylow $p$-subgroups. See \cite[Chapter~1]{Ser}.) Therefore if $E=F(\root{p}\of{a})$ is a cyclic extension of $F$ of degree $p$, so is $EL=L(\root{p}\of{a})$ over $L$. By Theorem~\ref{th:es} we see that the sequence \begin{equation*} k_{m-1}EL\xrightarrow{N_{EL/L}} k_{m-1}L \xrightarrow{\{a\}\cdot-} k_m L\xrightarrow{i_{EL}} k_m EL \end{equation*} is exact for each $m\in\N$. We claim that $i_L\colon k_mF \to k_mL$ is injective. Indeed, suppose that $i_L(\alpha)=0$ for some $\alpha\in k_m F$. Then there exists a finite subextension $M/F$ of $L/F$ such that $i_M(\alpha)=0$. Then \begin{equation*} 0=N_{M/F}(i_M(\alpha))=[M:F]\alpha, \end{equation*} (see \cite[p.~300]{FV}). Because $[M:F]$ is coprime with $p$, we see that $\alpha=0$ and $i_L$ is injective as asserted. Similarly we have that $i_{EL}\colon k_m E\to k_m EL$ is injective. We then have the following commutative diagram: \begin{equation*} \xymatrix{ k_{m-1}EL \ar[r]^{N_{EL/L}} & k_{m-1}L \ar[r]^{\{a\}\cdot -} & k_m L \ar[r]^{i_{LE}} & k_m EL \\ k_{m-1}E \ar[r]^{N_{E/L}} \ar[u]_{i_{EL}} & k_{m-1}F \ar[r]^{\{a\}\cdot -} \ar[u]_{i_{L}} & k_m F \ar[r]^{i_E} \ar[u]_{i_{L}} & k_m E \ar[u]_{i_{EL}} \\ } \end{equation*} Because the vertical maps are injective, we see that the bottom row of the diagram is a complex: the composition of any two consecutive maps is the zero map. We now establish exactness at the second and third terms of the complex. Let $\alpha\in k_{m-1}F$ such that $\{a\}\cdot\alpha=0$. Then $\{a\}\cdot i_L(\alpha)=0$ and therefore there exists an element $\beta\in k_{m-1}EL$ such that $N_{EL/L}(\beta)=i_L(\alpha)$. Let $M/F$ be a finite extension such that $\beta$ is defined over $EM$. Then $N_{EM/M}(\beta)=i_M(\alpha)$, and we have \begin{equation*} N_{EM/F}(\beta) = N_{M/F}(N_{EM/M}(\beta)) = N_{M/F}(i_M (\alpha)) = [M:F]\alpha \end{equation*} and $N_{EM/F}(\beta)=N_{E/F}(N_{EM/E}(\beta))$. Thus \begin{equation*} N_{E/F}(N_{EM/E}(\beta))=[M:F]\alpha. \end{equation*} Because $([M:F],p)=1$ we see that $\alpha\in N_{E/F}(k_{m-1}E)$. Therefore we have established the exactness of our complex at $k_{m-1}F$. Now assume that $\alpha\in k_m F$ such that $i_E(\alpha)=0\in k_m E$. Then arguing as above, we see that there exist a finite extension $M/F$ and $\beta\in k_{m-1}M$ such that \begin{equation*} \{a\}\cdot \beta = i_M(\alpha)\in k_m M. \end{equation*} Applying $N_{M/F}$ and using the projection formula we see that \begin{equation*} \{a\}\cdot N_{M/F}(\beta) = N_{M/F}(i_M(\alpha)) = [M:F]\alpha. \end{equation*} Because $[M:F]$ is coprime with $p$, $\alpha\in\{a\}\cdot N_{M/F}(\gamma)$ for a suitable element $\gamma\in k_{m-1}M$. Hence we see that our complex is also exact at $k_m F$ and the full complex is exact. \qed \begin{remark*} Assuming as usual that $F$ contains a primitive root $\xi_p$, then if $m\leq 2$, no further assumption on $F$ in Theorem~\ref{th:esext} is necessary. For $m=1$ this claim follows from basic Kummer theory, and for $m=2$ see \cite[Prop.~5]{Me} and \cite[Chap.~5, Lemma~8.4]{Sr}. \end{remark*} \section{Reduction to the Case $\chr F = 0$}\label{Red} Suppose now that $\chr F = q > 0$ and $q\neq p$. We also assume that $F$ is infinite, because if $F$ is finite then $K_n F=0$ for $n\geq 2$ and therefore this is a trivial case. (See \cite[Prop.~IX.1.3]{FV}.) Assume as before that $F$ contains a primitive $p$th root of unity $\xi_p$ and $E/F$ is a cyclic extension of degree $p$. We shall show that there exists an explicitly defined cyclic extension $J/L$ of degree $p$ such that $\chr L = 0$, so that $L$ is perfect, and $k_n J$ is naturally isomorphic with $k_n E$ as a $G=\Gal(E/F) \cong \Gal(J/L)$ module. Recall first that there exists a discrete valuation ring $A$ of characteristic $0$ such that its maximal ideal $M$ is generated by $q$ and $A/M\cong F$. (Such a ring is called a $q$-ring. See \cite[p.~223]{Ma}.) By passing to a completion $\hat A$ of $A$ with respect to $M$-adic topology and observing that $\hat A/\hat M\cong A/M\cong F$ and $\hat M=\hat A.q$ we see that we may and will assume that $A$ is a complete local $q$-ring such that $A/M\cong F$. It is known that a complete $q$-ring is uniquely determined up to its isomorphism by its residue field. (See \cite[Cor., p.~225]{Ma}.) Observe further that a complete discrete valued field is henselian. (See \cite[Thm.~5]{R}.) Now following \cite[\S IX.3]{FW} we have a natural construction \begin{align*} R &=\varinjlim(A^{(1)}\subset A^{(2)}\subset A^{(3)}\subset\cdots), \mbox{ where } \\ A^{(1)} &=A \mbox{ as above and } A^{(n+1)} := A^{(n)}[t]/(t^p-\pi_n), \end{align*} where $\pi_n$ is a uniformizer of $A^{(n)}$ for $n\ge 1$. As was noticed in \cite{FW}, this ring $R$ is a henselian valuation ring of characteristic $0$, with value group the underlying group of the ring $\Z[1/p]$. For each $i\in\N$ let $L^{(i)}$ be the quotient field of $A^{(i)}$, and let $L$ be the quotient field of $R$. Then from Lemma~IX.3.5 in \cite{FW}, there is a natural isomorphism $k_n F\cong k_nL$ for each $n\in\N$. Let $T$ be the inertia subgroup of $G_{L^{(1)}}$. We have the natural isomorphism $G_{L^{(1)}}/T\xrightarrow{\thicksim} G_F$. Let $\varphi$ be the compositum of the natural surjections \begin{equation*} G_{L^{(1)}}\to G_{L^{(1)}}/T \xrightarrow{\thicksim} G_F \to \Gal(E/F), \end{equation*} and let $J^{(1)}$ be the fixed field of $\ker\varphi$. Then $\Gal(J^{(1)}/L^{(1)})$ is naturally isomorphic to $\Gal(E/F)$. (This is a special case of a more general construction about lifting certain Galois abelian extensions. See \cite[Lemma~2.5]{K}.) Thus we see that $J^{(1)}/L^{(1)}$ is a cyclic, purely inert extension of degree $p$. Because the tower \begin{equation*} L^{(1)}\subset L^{(2)}\subset\dots\subset L^{(n)}\subset \cdots \end{equation*} is a chain of totally ramified extensions $L^{(n+1)}/L^{(n)}$, we see that $J^{(1)}\cap L=L^{(1)}$. Set $J:=J^{(1)} L$. Then $J/L$ is a cyclic, purely inert extension of degree $p$. Now set $J^{(i)}:=J^{(1)}L^{(i)}$, and let $B^{(i)}$ be the unique valuation ring in $J^{(i)}$ such that $B^{(i)}\cap L^{(i)}=A^{(i)}$. ($B^{(i)}$ is unique because $A^{(i)}$ is henselian.) Then again following the proofs of Lemma~IX.3.2 and Lemma~IX.3.5 in \cite{FW}, we establish, in exactly the same way we proved that $k_nF\cong k_nL$, that $k_n E\cong k_n J$ under a $G$-equivariant isomorphism. Our reduction is complete. \section{Acknowledgements} We are very grateful to M.~Rost and to J.-P. Serre for correspondence concerning the Bloch-Kato conjecture. We are also very grateful to J.-P.~Tignol for his kind and encouraging remarks concerning this paper. \begin{thebibliography}{MeSu} \bibitem[A]{A} J.~Kr.~Arason. Cohomologische invarianten quadratischer Formen. \textit{J.~Algebra} {\bf 36} (1975), 448--491. \bibitem[B]{B} Z.~I.~Borevi\v{c}. The multiplicative group of cyclic $p$-extensions of a local field. \textit{Trudy Mat.~Inst.~Steklov} \textbf{80} (1965), 16--29. Translated as \textit{Proc.~Steklov Inst.~Math.~No.~80 (1965): Algebraic number theory and representations}, edited by D.~K.~Faddeev, 15--30. Providence, RI: American Mathematical Society, 1968. \bibitem[FV]{FV} I.~B.~Fesenko and S.~V.~Vostokov. \textit{Local fields and their extensions}, 2nd ed. Translations of Mathematical Monographs 121. Providence, RI: American Mathematical Society, 2002. \bibitem[FW]{FW} E.~M.~Friedlander and C.~W.~Weibel. An overview of algebraic K-theory. \textit{Algebraic $K$-theory and its applications (Trieste, 1997)}, edited by H.~Bass, A.~O.~Kuku, and C.~Pedrini, 1--119. River Edge, NJ: World Sci.~Publishing, 1999. \bibitem[K]{K} J.~Koenigsmann. Encoding valuations in absolute Galois groups. \textit{Proceedings of the international conference and workshop on Valuation Theory (Saskatoon, 1999)}, edited by F.-V.~Kuhlmann, S.~Kuhlmann, and M.~Marshall, 107--132. Fields Institute Communications 33. Providence, RI: American Mathematical Society, 2003. \bibitem[Ma]{Ma} H.~Matsumura. \textit{Commutative ring theory}, 2nd ed. Cambridge Studies in Advanced Mathematics 8. Cambridge: Cambridge University Press, 1989. \bibitem[Me]{Me} A.~S.~Merkurjev. $K_2$ of fields and the Brauer group. \textit{Applications of algebraic $K$-theory to algebraic geometry and number theory (Boulder, Colo., 1983)}, Part II, 529--546. Contemporary Mathematics 55. Providence, RI: American Mathematical Society, 1986. \bibitem[MeSu]{MeSu} A.~S.~Merkurjev and A.~A.~Suslin. $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism. \textit{Izv.~Akad.~Nauk~SSSR Ser.~Mat.} {\bf 46} (1982), no.~5, 1011--1046, 1135--1136. Translated as \textit{Math. USSR-Izv.} {\bf 21} (1983), no.~2, 307--340. \bibitem[Mi]{Mi} J.~Milnor. Algebraic $K$-theory and quadratic forms. \textit{Invent.~Math.} {\bf 9} (1970), 318--344. \bibitem[MS]{MS} J.~Min\'a\v{c} and J.~Swallow. Galois module structure of $p$th-power classes of extensions of degree $p$. \textit{Israel J.~Math.} {\bf 138} (2003), 29--42. \bibitem[R]{R} P.~Ribenboim. Equivalent forms of Hensel's lemma. \textit{Exposition.~Math.} {\bf 3} (1985), no.~1, 3--24. \bibitem[Se]{Ser} J.-P.~Serre. \textit{Galois cohomology}. Berlin: Springer-Verlag, 1997. English translation, with revisions, of \textit{Cohomologie Galoisienne}, Lecture Notes in Math.~5, Springer-Verlag, 1964. \bibitem[Sr]{Sr} V.~Srinivas. \textit{Algebraic $K$-theory}, 2nd ed. Progress in Mathematics 90. Boston: Birkh\"auser Boston, 1996. \bibitem[V1]{Vo1} V.~Voevodsky. On $2$-torsion in motivic cohomology. arXiv:math.AG \linebreak /0107110. Available: http://arXiv.org/abs/math/0107110 (2001). \bibitem[V2]{Vo2} V.~Voevodsky. On motivic cohomology with $\Z/l$ coefficients. K-theory preprint archive 639. Available: http://www.math.uiuc.edu/K-theory/ \linebreak 0639/ (2003). \end{thebibliography} \end{document}