Computational methods in constructive Galois theory

Abstract

This lecture can be viewed as a complement to my lecture [18] given in Berkeley. It begins with a short survey of the known rationality criteria for Galois extensions over ℂ(t1,...,ts) or, equivalently, for Galois coverings of the projective space ℙS(ℂ). In the subsequent sections some computational problems are discussed, which arise in the application of these theorems: computation of class numbers of generators of finite groups, computation of the braid orbits on classes of generators, construction of polynomials with given ramification structure, determination of Galois groups. The computational methods are examplified by the Mathieu groups M11,...,M24. So polynomials with Galois group M11,M12,M22 over ℚ are constructed and the existence of Galois extensions with Galois group M24 over ℚ is proved.