v-Adic maximal extensions, spectral norms and absolute Galois groups

Abstract

Let (K, v) be a perfect rank one valued field and let ${(\overline{K_{v}},\overline{v})}$ be the canonical valued field obtained from (K, v) by the unique extension of the valuation ${\widetilde{v}}$ of K _v , the completion of K relative to v, to a fixed algebraic closure ${\overline{K_{v}}}$ of K _v . Let ${\overline{K}}$ be the algebraic closure of K in ${\overline {K_{v}}}$. An algebraic extension L of K, ${L\subset\overline{K}}$, is said to be a v-adic maximal extension, if ${\overline{v}\mid_{L}}$ is the only extension of v to L and L is maximal with this property. In this paper we describe some basic properties of such extensions and we consider them in connection with the v-adic spectral norm on ${\overline{K}}$ and with the absolute Galois groups Gal${(\overline{K}/K)}$ and Gal${(\overline{K_{v}} /K_{v})}$. Some other auxiliary results are given, which may be useful for other purposes.