Partial Results
There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any function field in one variable over the complex numbers C, and more generally over function fields in one variable over any algebraically closed field of characteristic zero. Shafarevich showed that every finite solvable group is realizable over Q. It is also known that every sporadic group, except possibly the Mathieu group M23, is realizable over Q.
Hilbert had shown that this question is related to a rationality question for G: if K is any extension of Q, on which G acts as an automorphism group and the invariant field KG is rational over Q, then G is realizable over Q. Here rational means that it is a purely transcendental extension of Q, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric groups are realizable.
Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing G geometrically as a Galois covering of the projective line: in algebraic terms, starting with an extension of the field Q(t) of rational functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group.
Read more about this topic: Inverse Galois Problem
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