Subfields of algebraically maximal Kaplansky fields

TL;DR

This paper explores the structure of algebraically maximal Kaplansky fields, demonstrating that they contain maximal immediate extensions of their subfields, with implications for henselian valued fields of higher rank.

Contribution

It introduces new results on the subfield extension properties of algebraically maximal Kaplansky fields using ramification theory.

Findings

Maximal Kaplansky fields contain maximal immediate extensions of their subfields.

Algebraically maximal Kaplansky fields contain maximal immediate algebraic extensions.

Results have implications for henselian valued fields of rank higher than 1.

Abstract

Using the ramification theory of tame and Kaplansky fields, we show that maximal Kaplansky fields contain maximal immediate extensions of each of their subfields. Likewise, algebraically maximal Kaplansky fields contain maximal immediate algebraic extensions of each of their subfields. This study is inspired by problems that appear in henselian valued fields of rank higher than 1 when a Hensel root of a polynomial is approximated by the elements generated by a (transfinite) Newton algorithm.