Russell Miller: Computability questions about infinite Galois groups
Автор: Hausdorff Center for Mathematics
Загружено: 2026-01-06
Просмотров: 123
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For an infinite algebraic field extension $E/F$, Calvert, Harizanov, and Shlapentokh described the automorphisms of $E$ over $F$ as the set of paths through a tree. Under some basic computability assumptions about $E$ and $F$, this tree is computable, and composition and inversion of the paths are both computable by Turing functionals. Thus we have an effective presentation of the Galois group $\operatorname{Gal}(E/F)$, despite its potentially-continuum size.
For finite fields, this presentation of the absolute Galois group is extremely nice, indeed as close to being a decidable structure as one can get in this cardinality. We will explain the notion of ``tree-decidability,'' developed by Block and the speaker, that makes this rigorous. Predictably, the absolute Galois group of $\mathbb Q$ is nastier: we will quantify this nastiness in a few specific ways -- one of them joint with Kundu -- and pose several open questions about exactly how bad it gets.
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