The determination of Galois groups

Abstract

A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the polynomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear factors of which determine new orderings for the roots. Sequences of these resolvents isolate the Galois group of the polynomial. Machine implementation of the technique requires the use of multiple-precision integer and multiple-precision real and complex floating-point arithmetic. Using this technique, the writer has developed programs for the determination of the Galois groups of polynomials of degree N ≦ 7 N \leqq 7 . Two exemplary calculations are given.