Absolute Galois group, number theory and the Axiom of Choice Richard Taylor once explained his research (in number theory) in a very simple way: understanding the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
It is known that in Zermelo-Fraenkel theory without the Axiom of Choice, it is consistent to say that $\vert \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \vert = 2$.
What are the consquences for number theory in such a model ?
 A: In the absence of the axiom of choice, it is still possible to define the "usual" algebraic closure of $\mathbb{Q}$ because you can just explicitly enumerate all polynomials with integer coefficients.  Richard Taylor's remark should be interpreted as referring to this particular algebraic closure.
What you can't prove without choice is the uniqueness of the algebraic closure. Exotic algebraic closures could exist. So in the absence of choice, it's a bit misleading to use the notation $\overline{\mathbb{Q}}$ as if it referred to some unique thing. Wilfrid Hodges, in his paper, Läuchli's algebraic closure of $Q$, shows that an exotic algebraic closure of $\mathbb{Q}$ can have all kinds of unexpected properties.  But what this shows, in my opinion, is that what we usually think of as "algebraic number theory" doesn't make too much sense in that context.  When people talk about "algebraic number theory" they are tacitly referring to the usual algebraic closure.
A: There would be no consequences, for two reasons:

*

*As Timothy Chow points out, if we define $\overline{\mathbb Q}$ as the set of complex numbers that are roots of a nonzero polynomial with rational coefficients, it is easy to prove the Galois group maps surjectively to the automorphism group of each finite Galois extension of $\mathbb Q$.


*If we wanted to consider the Galois group of another field where there there really are set-theoretic issues with finding enough automorphisms (e.g. studying the automorphism group of $\mathbb C$) we could simply redefine our notion of "group" so it doesn't necessarily have a set of elements (e.g. as a group object in the category of locales) and there would be no difficulty with redoing the standard arguments in this setting.
