<Emphasis Type="Italic">v</Emphasis>-Adic maximal extensions,spectral norms and absolute Galois groups

Let (K, v) be a perfect rank one valued field and let (`(K<sub>v</sub>)],`(v)]){(\overline{K_{v}},\overline{v})} be the canonical valued field obtained from (K, v) by the unique extension of the valuation (v)\tilde]{\widetilde{v}} of K _v , the completion of K relative to v, to a fixed algebraic closure `(K<sub>v</sub>)]{\overline{K_{v}}} of K <sub>v</sub> . Let `(K)]{\overline{K}} be the algebraic closure of K in `(K<sub>v</sub>)]{\overline {K_{v}}}. An algebraic extension L of K, L ì `(K)]{L\subset\overline{K}}, is said to be a v-adic maximal extension, if `(v)] | <sub>L</sub>{\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 `(K)]{\overline{K}} and with the absolute Galois groups Gal(`(K)]/K){(\overline{K}/K)} and Gal(`(K_v)] /K_v){(\overline{K_{v}} /K_{v})}. Some other auxiliary results are given, which may be useful for other purposes.