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The Embedding Problem in Galois Theory

B. B. Lur′e Academy of Sciences of the USSR
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Hardcover ISBN: 978-0-8218-4592-9
Product Code: MMONO/165
List Price: $110.00 MAA Member Price:$99.00
AMS Member Price: $88.00 Electronic ISBN: 978-1-4704-4580-5 Product Code: MMONO/165.E List Price:$103.00
MAA Member Price: $92.70 AMS Member Price:$82.40
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List Price: $165.00 MAA Member Price:$148.50
AMS Member Price: $132.00 Click above image for expanded view The Embedding Problem in Galois Theory B. B. Lur′e Academy of Sciences of the USSR D. K. Faddeev Academy of Sciences of the USSR Available Formats:  Hardcover ISBN: 978-0-8218-4592-9 Product Code: MMONO/165  List Price:$110.00 MAA Member Price: $99.00 AMS Member Price:$88.00
 Electronic ISBN: 978-1-4704-4580-5 Product Code: MMONO/165.E
 List Price: $103.00 MAA Member Price:$92.70 AMS Member Price: $82.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$165.00 MAA Member Price: $148.50 AMS Member Price:$132.00
• Book Details

Translations of Mathematical Monographs
Volume: 1651997; 182 pp
MSC: 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.

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 non-Abelian 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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Volume: 1651997; 182 pp
MSC: 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.

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 non-Abelian 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
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