Automorphism towers and automorphism groups of fields without Choice

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.