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

Russell Miller

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arXiv · 2307.08935

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

Russell Miller

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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.

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Russell Miller. 2023-07-18. Computability for the absolute Galois group of $\mathbb{Q}$. https://arxiv.org/abs/2307.08935

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