# Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?)

I suspect that the theory of Galois groups is not a first-order theory, because otherwise we could apply Loewenheim-Skolem's theorem, since there exists an infinite Galois group (e.g. $\hat{\mathbb Z}$), and thus obtain a countable model, i.e. a countable Galois group. So I can't apply anything of the model theory I know (i. e. only first-order).

Maybe an ultraproduct construction could help? Thank you very much.

• that's the first time i've heard of a "theory of galois groups". what is the language it should be using and what is it supposed to say ? – mercio May 18 '16 at 12:53
• It should be more simple to figure out what the "profinite group theory" is supposed to say. In that case the language should be the same of the group theory, and we should add such properties as compactness, T2, and having a fundamental neighborhhod system of $1$ made by normal subgroups. It is not surprising that these properties sound like higher-order properties. – W. Rether May 18 '16 at 13:21