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.

Autorentext

Education: PhD in Mathematics, The University ofWestern Ontario. Current Position: Assistant Professor atDepartment of Mathematics, Tarbiat Modares University, Tehran.Research Interest: Noncommutative Geometry, OperatorAlgebras and Galois Theory.

Klappentext

This book is concerned with a type of embeddingproblems in Galois theory known as Galois embeddingproblems with finite abelian kernels of exponent pover cyclic Galois extensions E/F whose Galoisgroups are of exponent p and F contains a primitivep-th root of unity. We develop a method to describeall solutions of these problems and apply our methodin a special case. Our constructive approach providesthe theoretical background for an explicitcomputation of all Kummer extensions of exponent pover E which are Galois over F too. In Chapter 1, werecollect some basic concepts of Galois theory,inverse Galois theory, Galois embedding problems andinfinite Galois theory. We also explain the role ofT-groups in infinite Galois theory. Chapter 2 isdevoted to a constructive study of the module theorynecessary for our work and a theoretical framework tostudy homomorphism between two finitely generatedmodules in terms of linear algebra. A relativeversion of Kummer theory is developed in Chapter 3.This book is concluded with a study of differentaspects of our Galois embedding problems in Chapter 4.