Valuation bases for generalized algebraic series fields

TL;DR

This paper studies valued fields with valuation bases, providing conditions under which subfields of generalized power series fields admit such bases, enabling the construction of real closed fields with exponential functions.

Contribution

It introduces a sufficient condition for valued subfields of generalized power series fields to admit valuation bases, applicable to rational functions and algebraic power series.

Findings

F(G) admits a valuation basis under certain conditions

F(G) can be equipped with a restricted exponential function

Results apply to archimedean F and divisible G

Abstract

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F, we give a sufficient condition for a valued subfield of the field of generalized power series F((G)) to admit a K-valuation basis. We show that the field of rational functions F(G) and the field F(G) of power series in F((G)) algebraic over F(G) satisfy this condition. It follows that for archimedean F and divisible G the real closed field F(G) admits a restricted exponential function.