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

摘要

Let (K, nu) be a perfect rank one valued field and let ((K-nu) over bar,(nu) over bar) be the canonical valued field obtained from ( K, nu) by the unique extension of the valuation (nu) over tilde of K-nu, the completion of K relative to nu, to a fixed algebraic closure (K-nu) over bar of K-nu. Let (K) over bar be the algebraic closure of K in (K-nu) over bar. An algebraic extension L of K, L subset of (K) over bar, is said to be a nu-adic maximal extension, if (nu) over bar vertical bar(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 nu-adic spectral norm on (K) over bar and with the absolute Galois groups Gal((K) over bar /K) and Gal((K-nu) over bar /K-nu). Some other auxiliary results are given, which may be useful for other purposes.