Victor Lisinski: Hilbert's Irreducibility Theorem and Applications to the Inverse Galois Problem

Abstract:

We will explore a powerful and very useful result in Number Theory called Hillbert's Irreducibility Theorem. In its most basic form, this theorem states that for any irreducible polynomial P(T,X) with coefficients in the field of rational functions over Q, there is always an element t in Q for which the polynomial P(t,X) with coefficients in Q is irreducible (i.e. the polynomial obtained by evaluating the coefficients of P at t is still irreducible). We will apply this theorem to obtain some fundamental results regarding the still unsolved question if all finite groups appear as Galois groups of some Galois extension K/Q. It turns out that Hilbert's Irreducibility Theorem can reduce this problem to the question whether or not every finite group is realizable as a Galois group of some Galois extension of Q(T1,...,Tn). Finally, we show that the alternating group has this property.