  The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the
field $\mathbb{Q}$ of rational numbers can be presented as a highly computable
object, under the notion of type-2 Turing computation. We formalize such a
presentation and use it to address several effectiveness questions about
Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem
functions for this group, the arithmetical complexity of various definable
subsets of the group, and the extent to which countable subgroups defined by
complexity (such as the group of all computable automorphisms of the algebraic
closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall
group.
