Galois Embedding Problems

This book is concerned with a type of embedding problems in Galois theory known as Galois embedding problems with finite abelian kernels of exponent p over cyclic Galois extensions E/F whose Galois groups are of exponent p and F contains a primitive p-th root of unity. We develop a method to describe all solutions of these problems and apply our method in a special case. Our constructive approach provides the theoretical background for an explicit computation of all Kummer extensions of exponent p over E which are Galois over F too. In Chapter 1, we recollect some basic concepts of Galois theory, inverse Galois theory, Galois embedding problems and infinite Galois theory. We also explain the role of T-groups in infinite Galois theory. Chapter 2 is devoted to a constructive study of the module theory necessary for our work and a theoretical framework to study homomorphism between two finitely generated modules in terms of linear algebra. A relative version of Kummer theory is developed in Chapter 3. This book is concluded with a study of different aspects of our Galois embedding problems in Chapter 4.