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.
