Graded transcendental extensions of graded fields

M Boulagouaz

doi:10.1155/s0161171203210656

Abstract

We study transcendency properties for graded field extension and give an application to valued field extensions.

Highlights

An important tool to study rings with valuation is the socalled associated graded ring construction: to a valuation ring R, we can associate a ring gr(R) graded by the valuation group
This note is a continuation of earlier work of the author, in which graded fields and graded division rings are studied with special emphasis on applications to valuation theory
The aim of this note is to introduce and study the notion of gr-transcendental graded field extension, at least in the case where the grading group is torsion-free abelian; application to valued field extensions leads to three different notions of transcendental extensions of valued fields

Summary

Introduction

An important tool to study rings with valuation is the socalled associated graded ring construction: to a valuation ring R, we can associate a ring gr(R) graded by the valuation group. Every graded field extension of finite degree is gr-algebraic (see [6] or [7]); [S : R] = ∞ if R ⊂ S is transcendental. We call S a pure gr-transcendental graded field extension of R if there exists a (possibly empty) gr-transcendency basis T of R such that S = R(T ).

Results

Conclusion