Computing Galois Groups

Abstract

In this chapter, we discuss computations of Galois groups. In general, computing the Galois group of a given polynomial is numerically complicated when the degree of the polynomial is modestly high. The numerical methods depend on the knowledge of transitive subgroups of the symmetric groups. Here, we discuss some theoretical background on numerical methods (which are implemented in some computer packages) and apply it in a few cases. In the exercises, we illustrate how to compute and classify Galois groups for low degree polynomials by specifying some numerical invariants, which provides information on the isomorphism type of the Galois group depending on their values. We do this for polynomials of degrees 3 and 4. We further discuss the Galois resolvents and use them to proof a general theorem by Richard Dedekind, which relates the Galois group of an integer irreducible polynomial to Galois groups of its reductions modulo prime numbers. Several exercises are concerned with Dedekind’s theorem, allowing for the construction of polynomials with given Galois groups and the solution of the inverse problem for the symmetric group S_n.