Computation of Galois groups over function fields

Abstract

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q ( t 1 , t 2 , … , t m ) \mathbb {Q}(t_1,t_2,\ldots ,t_m) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.