Little galoisian modules

Abstract

Let $p$ be a prime number, let $K$ be a $p$-field (a local field with finite residue field of characteristic $p$), let $L$ be a finite galoisian tamely ramified extension of $K$, and let $G=\mathrm{Gal}(L|K)$. Suppose that $L$ is split over $K$ in the sense that the short exact sequence $1\to T\to G\to G/T\to1$ has a section, where $T$ is the inertia subgroup of $G$. We determine the structure of the $\mathbf{F}_p[G]$-module $L^\times\!/L^{\times p}$ in characteristic $0$ when the $p$-torsion subgroup ${}_pL^\times$ of $L^\times$ has order $p$, and of the $\mathbf{F}_p[G]$-modules $L^\times\!/L^{\times p}$ and $L^+\!/\wp(L^+)$ in characteristic $p$, where $\wp(x)=x^p-x$. Let $\tilde K$ be a maximal galoisian extension of $K$, let $V$ be the maximal tamely ramified extension of $K$ in $\tilde K$, let $\Gamma=\mathrm{Gal}(V|K)$, and let $B$ be the maximal abelian extension of exponent $p$ of $V$ in $\tilde K$. We determine the structure of the $\mathbf{F}_p[[\Gamma]]$-module $\mathrm{Gal}(B|V)$, and show how this leads in characteristic $0$ to a simple proof of the fact that the profinite group $\mathrm{Gal}(\tilde K|K)$ is generated by $[K:\mathbf{Q}_p]+3$ elements.