Automorphism towers and automorphism groups of fields without Choice

TL;DR

This paper explores automorphism towers and groups of fields without the Axiom of Choice, introducing new connections, proofs, and graph-theoretic methods to represent groups as automorphism groups of fields.

Contribution

It establishes a link between automorphism and normalizer towers without Choice and provides a new graph-based proof of a theorem representing any group as a field automorphism group.

Findings

Connected automorphism and normalizer towers without Choice.

New proof that any group can be realized as a field automorphism group.

Utilized graph theory to simplify the construction.

Abstract

This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower and the normalizer tower without Choice and introduce a new proof to a theorem of Fried and Koll\'ar that any group can be represented as an automorphism group of a field. The proof uses a simple construction: working more in graph theory, and less in algebra.