The Inverse Galois Problem: an Overview

The inverse Galois problem consists of determining whether a finite group G occurs as a Galois group over some field K, that is, whether a Galois extension L/K exists such that Gal(L/K) is isomorphic to G. The classical version of the problem deals with the existence issue for the field K=Q of rational numbers: an ongoing dilemma still relevant nowadays. An overview is presented: in this work, some of the milestones achieved in the last two centuries find place: from the Kronecker-Weber result, concerning the realisation problem for abelian groups, to Hilbert's irreducibility theorem, a first approach of the so-called methods of descent, whom address the realisation of G as a Galois over some regular extension Q(x_1,...,x_m) of Q and take advantage of its invariance under the process that sees the evaluation of the transcendentals x_1,...,x_m. The latter had suggested other techniques, such as the rigidity criteria, which primarily handle finite extensions of F(x) (F an algebraically closed subfield of C), with the purpose of finding conditions to result in Q: it will be shown that a finite group G with Q-rational classes describing a rigid tuple, occurs as a Galois over Q. Some applications will also be displayed: from the classical ones, like the realisation of S_n, A_n or some sporadic groups, to the more modern ones, included the observation of the Rubik's cube group regarded as a Galois group.