Topics in Inverse Galois Theory
Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q.