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 $1\to T\to G\to G/T\to 1$ 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 ${}_p{L}^{\times }$ 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.

References in corpus (1)

• Evolution of proto-galaxy-clusters to their present form: theory and observations

Cited by in corpus (2)

• Wildly primitive extensions
• Solvable primitive extensions