Hilbert's irreducibility theorem and its application to the inverse Galois problem

Abstract

To every polynomial f (x) with rational coefficients one can associate a finite group Gf , the Galois group of the splitting field of f over the rational numbers. The inverse problem of Galois theory asks whether for a given finite group G, there exists a polynomial f such that G is isomorphic to Gf. A Galois extension of Q, with Galois group G, is called a realisation of G over Q, and G is said to occur over Q. It is known that all abelian groups occur over Q, and Šafereviè showed in 1957 that all solvable groups occur over Q. Almost all other progress with the problem depends on Hilbert’s irreducibility theorem, which implies that a realisation of G over Q exists if and only if a realisation exists over the function field Q (x). Hence it suffices to find realisations of a particular group G over Q (x), which enables us to use tools from Riemannian Surface Theory and Algebraic Geometry.