Subfields of Valued Field and Galois Theory of Transcendental Extensions

Abstract

This Dissertation Focuses on: 1. Exploring the Subfields of a higher rank valued field (K,v), which are closed (Complete) under the topology induced by the valuation v on K. By following this direction, we generalize some known results from classical theory of valuations [[18], [13], or [19] for structure theorems of local fields]. 2. We introduce, by following some ideas of O. F. G. Schilling [16], a new construction for a Galois theory of a class of subextensions of k((X))/k, This will help us to obtain new information on the structure of the subfields L of k((X)) and, implicitly, on some transcendental functions of k((X)).