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Book DetailsTranslations of Mathematical MonographsVolume: 165; 1997; 182 ppMSC: Primary 12; Secondary 11;
The central problem of modern Galois theory involves the inverse problem: given a field \(k\) and a group \(G\), construct an extension \(L/k\) with Galois group \(G\). The embedding problem for fields generalizes the inverse problem and consists in finding the conditions under which one can construct a field \(L\) normal over \(k\), with group \(G\), such that \(L\) extends a given normal extension \(K/k\) with Galois group \(G/A\). Moreover, the requirements applied to the object \(L\) to be found are usually weakened: it is not necessary for \(L\) to be a field, but \(L\) must be a Galois algebra over the field \(k\), with group \(G\). In this setting the embedding problem is rich in content. But the inverse problem in terms of Galois algebras is poor in content because a Galois algebra providing a solution of the inverse problem always exists and may be easily constructed. The embedding problem is a fruitful approach to the solution of the inverse problem in Galois theory.
This book is based on D. K. Faddeev's lectures on embedding theory at St. Petersburg University and contains the main results on the embedding problem. All stages of development are presented in a methodical and unified manner.ReadershipGraduate students and research mathematicians interested in field theory and polynomials.

Table of Contents

Chapters

Chapter 1. Preliminary information about the embedding problem

Chapter 2. The compatibility condition

Chapter 3. The embedding problem with Abelian kernel

Chapter 4. The embedding problem for local fields

Chapter 5. The embedding problem with nonAbelian kernel for algebraic number fields

Appendix


Reviews

The English translation is particularly welcome because it contains a full and simplified proof of the existence theorem of Shafarevich for normal extensions of an algebraic number field with given solvable Galois group.
Zentralblatt MATH


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The central problem of modern Galois theory involves the inverse problem: given a field \(k\) and a group \(G\), construct an extension \(L/k\) with Galois group \(G\). The embedding problem for fields generalizes the inverse problem and consists in finding the conditions under which one can construct a field \(L\) normal over \(k\), with group \(G\), such that \(L\) extends a given normal extension \(K/k\) with Galois group \(G/A\). Moreover, the requirements applied to the object \(L\) to be found are usually weakened: it is not necessary for \(L\) to be a field, but \(L\) must be a Galois algebra over the field \(k\), with group \(G\). In this setting the embedding problem is rich in content. But the inverse problem in terms of Galois algebras is poor in content because a Galois algebra providing a solution of the inverse problem always exists and may be easily constructed. The embedding problem is a fruitful approach to the solution of the inverse problem in Galois theory.
This book is based on D. K. Faddeev's lectures on embedding theory at St. Petersburg University and contains the main results on the embedding problem. All stages of development are presented in a methodical and unified manner.
Graduate students and research mathematicians interested in field theory and polynomials.

Chapters

Chapter 1. Preliminary information about the embedding problem

Chapter 2. The compatibility condition

Chapter 3. The embedding problem with Abelian kernel

Chapter 4. The embedding problem for local fields

Chapter 5. The embedding problem with nonAbelian kernel for algebraic number fields

Appendix

The English translation is particularly welcome because it contains a full and simplified proof of the existence theorem of Shafarevich for normal extensions of an algebraic number field with given solvable Galois group.
Zentralblatt MATH