Inverse Galois Problem

Abstract

In this thesis, motivated by the Inverse Galois Problem, we prove the occurence of Sn as Galois group over any global field. While Hilbert’s Irreducibility Theorem, the main ingredient of this proof, can be proved(for Q) using elementary methods of complex analysis, we do not follow this approach. We give a general form of Hilbert’s Irreducibility Theorem which says that all global fields are Hilbertian. Proving this takes us to Riemann hypothesis for curves and Chebotarev Density Theorem for function fields. In addition we prove the Chebotarev Density Theorem for Number Fields. The main reference for this thesis is [1] and the proofs are borrowed from the same.