#45: Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

MSRI Publications -- Volume 45

Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

By Christian U. Jensen, Arne Ledet, and Noriko Yui

This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of “generic” polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of “generic dimension” to address the problem of the smallest number of parameters required by a generic polynomial.