Computability for the absolute Galois group of $\mathbb{Q}$

TL;DR

This paper explores the computability of the absolute Galois group of the rationals, formalizing its structure within type-2 Turing computation to analyze various effectiveness and definability questions.

Contribution

It formalizes a computable presentation of the Galois group and investigates the complexity of definable subsets and subgroups within this framework.

Findings

Skolem functions for the group are computationally challenging.

Definable subsets have complex arithmetical properties.

Certain countable subgroups can be elementary subgroups of the entire group.

Abstract

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation and use it to address several effectiveness questions about Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem functions for this group, the arithmetical complexity of various definable subsets of the group, and the extent to which countable subgroups defined by complexity (such as the group of all computable automorphisms of the algebraic closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall group.