Henselian valued fields: a constructive point of view

Abstract

This article is a logical continuation of the Henri Lombardi and Franz-Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how (in the spirit of [9]) to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the valuation ring membership. It is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges (cf. [10]), who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)