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...
More
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” poly- nomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polyno- mials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of “generic dimen- sion” to address the problem of the smallest number of parameters required by a generic polynomial.
Less
