  Let $G$ be a finite group. Then there exists a first-order statement $S(G)$
in the language of rings without parameters and depending only on $G$ such
that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a
Galois extension with the Galois group isomorphic to $G$. Further, there is an
effective procedure which takes the table of multiplication of $G$ as its input
and produces $S_G$. Therefore, given a field $K$, the Inverse Galois Problem
for $K$, that is, the problem of deciding whether $K$ has a Galois extension
with a particular Galois group as input, is Turing reducible to the first-order
theory of $K$. Similar results hold for the Finite Split Embedding Problem and
the Inverse Automorphism Problem.
