Logic Seminar
Russell Miller
Queens College CUNY
Computability and elementary subgroups of 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 nicely 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 $\operatorname{Gal}(F)$. When $F=\mathbb Q$,
we have many questions and a few answers, partly due
to joint work with Debanjana Kundu.
In the simpler situations of the absolute Galois group of a finite field,
and of the Galois group of the cyclotomic field over $\mathbb Q$, much
more is known, thanks in part to joint work by Jason Block and the speaker.
Tuesday February 10, 2026 at 3:00 PM in 636 SEO