In this poster we show how the Fieker-Klueners algorithm together with the adjustments made for function fields can be used to compute Galois groups of reducible polynomials over function fields and fixed fields of subgroups of Galois groups computed using this algorithm.

Putting these two together we then describe how these Galois group computations can be used as part of a process which gives an explicit description of exceptions to Hilbert's Irreducibility Theorem. For a polynomial P in Q[t, x] with Galois group G over the field Q(t) these exceptions are rational numbers c such that the specialized polynomial P(c, x) has Galois group not isomorphic to G or factors differently from P.

The poster also describes an algorithm to compute the Geometric Galois group of a polynomial over Q(t). This algorithm avoids imprecise calculations in C by using an algebraic extension K of Q which contains the necessary algebraic numbers. It chooses the geometric Galois group from the subgroups of the Galois group of the polynomial using bounds on the indicies of the subgroups and the fact that K is an extension of Q.

The poster includes a description of the necessary algorithms as well as non-trivial examples showing the types of calculations that these algorithms make possible.