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 Kv, the completion of K relative to v, to a fixed algebraic closure \({\overline{K_{v}}}\) of Kv. 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.