information:classification, structural:summarization
Inverse Galois problem

The inverse Galois problem is to find a field extension with a given Galois group.

As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups.For showing this, one may proceed as follows. Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates { x }, one for each element α of G, and adjoin them to K to get the field F = K ({ x }). Contained within F is the field L of symmetric rational functions in the { x }. The Galois group of F / L is S, by a basic result of Emil Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F / M is G.

On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q. Various people have solved the inverse Galois problem for selected non-Abelian simple groups. Existence of solutions has been shown for all but possibly one (Mathieu group M) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group.