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.
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