Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions

Abstract

This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p{)}_E\ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$ does not possess nontrivial proper decomposition groups. When $r(p{)}_E=2$, it describes the decomposition groups of $\mathcal{G}(E(p)/E)$. The paper shows that if $(K,v)$ is a $p$-Henselian valued field with $r(p{)}_K\in \mathbb{N}$ and a residue field of characteristic $p$, then $P\cong \tilde{P}$ or $P$ is presentable as a semidirect product ${\mathbb{Z}}_p^{\tau }⋊\tilde{P}$, for some $\tau \in \mathbb{N}$, where $\tilde{P}$ is a Demushkin group of rank $\ge 3$ or a free pro-$p$-group. It also proves that when $\tilde{P}$ is of the former type, it is continuously isomorphic to $\mathcal{G}(K^{\prime }(p)/K^{\prime })$, for some local field $K^{\prime }$ containing a primitive $p$-th root of unity.

LaTex, 28 pages; an inaccuracy in the proof of Lemma 3.2 is fixed; improvements in the presentation at several other points

Cited by in corpus (1)

• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions