Use the fact that any finite group is isomorphic to a subgroup of to prove that given any finite group there exists fields and such that

Any field , are indeterminate. Let . is the field of symmetric rational functions in Or if we let are basic symmetric polynomials, namely

, , ,

then .

As we all know, . Since , by the Galois corresponding theorem, there exists an intermediate field such that .

This is the easiest inverse Galois problem since one does not specify the ground field. If you restrict , things are much trickier.