Free groups and automorphism groups of infinite fields

TL;DR

This paper constructs fields of large cardinality with automorphism groups that are free or free abelian groups, addressing a set-theoretic question about the existence of such automorphism groups.

Contribution

It demonstrates the existence of large fields with automorphism groups as free or free abelian groups under certain cardinal assumptions, and explores the set-theoretic implications.

Findings

Existence of fields with automorphism groups as free groups of size 2^λ.

Existence of fields with automorphism groups as free abelian groups of size 2^λ.

Set-theoretic conditions affecting the existence of such fields and automorphism groups.

Abstract

Let \lambda be a cardinal with \lambda=\lambda^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality \lambda and characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^\lambda and the automorphism group of K_1 is a free abelian group of cardinality 2^\lambda. This partially answers a question from [8] and complements results from [15], [16] and [17]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that the existence of a cardinal \lambda of uncountable cofinality with the property that there is no field of cardinality \lambda whose automorphism group is a free group of cardinality greater than \lambda implies the existence of large cardinals in certain inner models of set theory.