The fields of definition of branched Galois covers of the projective line

TL;DR

This paper investigates the fields of definition for Galois branched covers of the projective line, establishing minimal fields where automorphisms are defined and linking these to the inverse Galois problem.

Contribution

It proves the existence of minimal fields of definition for Galois covers and relates these fields to roots of unity and the inverse Galois problem.

Findings

Every cover model has a unique minimal field of definition.

Existence of a Galois field over the moduli field containing roots of unity.

Construction of number field extensions realizing Galois groups with controlled ramification.

Abstract

In this paper I explore the structure of the fields of definition of Galois branched covers of the projective line over \bar Q. The first main result states that every mere cover model has a unique minimal field of definition where its automorphisms are defined, and goes on to describe special properties of this field. One corollary of this result is that for every G-Galois branched cover there is a field of definition which is Galois over its field of moduli, with Galois group a subgroup of Aut(G). The second main theorem states that the field resulting by adjoining to the field of moduli all of the roots of unity whose order divides some power of |Z(G)| is a field of definition. By combining this result with results from an earlier paper, I prove corollaries related to the Inverse Galois Problem. For example, it allows me to prove that for every finite group G, there is an extension of number fields Q \subset E \subset F such that F/E is G-Galois, and E/Q ramifies only over those primes that divide |G|. I.e., G is realizable over a field that is "close" to Q.