Computability and absolute Galois groups

Abstract: The absolute Galois group $\operatorname{Gal}(F)$ of a field $F$ is the Galois group of its algebraic closure $\overline{F}$ relative to $F$, containing precisely those automorphisms of $\overline{F}$ that fix $F$ itself pointwise. Even for a field as simple as the rational numbers $\mathbb{Q}$, $\operatorname{Gal}(\mathbb Q)$ is a complicated object. Indeed (perhaps counterintuitively), $\operatorname{Gal}(\mathbb Q)$ is among the thorniest of all absolute Galois groups normally studied.

When $F$ is countable, $\operatorname{Gal}(F)$ usually has the cardinality of the continuum. However, it can be presented as the set of all paths through an $F$-computable finite-branching tree, built by a procedure uniform in $F$. We will first consider the basic properties of this tree, which depend in some part on $F$. Then we will address questions about the subgroup consisting of the computable paths through this tree, along with other subgroups similarly defined by Turing ideals. One naturally asks to what extent these are elementary subgroups of $\operatorname{Gal}(F)$ (or at least elementarily equivalent to $\operatorname{Gal}(F)$). This question is connected to the computability of Skolem functions for $\operatorname{Gal}(F)$, and also to the arithmetic complexity of definable subsets of $\operatorname{Gal}(F)$.

Some of the results that will appear represent joint work with Debanjana Kundu.

logicnumber theory

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic