A choice-free absolute Galois group and Artin motives

TL;DR

This paper constructs a canonical, choice-free version of the absolute Galois group as a profinite algebraic group, providing new insights into its structure and relation to Artin motives without relying on the axiom of choice.

Contribution

It introduces a canonical, inner form of the absolute Galois group that does not depend on the axiom of choice and relates it to Artin motives.

Findings

Constructed a choice-free, canonical profinite algebraic group as an inner form of the Galois group.

Defined a unique analogue of Frobenius elements in this new group.

Provided a new interpretation of the Galois group in the category of Artin motives.

Abstract

Proofs that an arbitrary field has a separable closure are necessarily non-constructive, and separable closures are unique only up to non-canonical isomorphism. This means that the absolute Galois group of a field is defined only up to inner automorphism. Here we construct a profinite algebraic group which is an inner form of the absolute Galois group. Our construction uses no form of the axiom of choice, and the group is defined up to canonical isomorphism. We also show that the Frobenius associated with a prime of a number field unramified in an extension, which is classically defined only up to conjugation, has a uniquely-defined analogue in terms of our group. We give a construction of the category of Artin motives with coefficients in an arbitrary field, and we give an interpretation of our absolute Galois group in terms of this category.