Ebook: Algebra: Fields with Structure, Algebras and Advanced Topics

The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.

The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.

The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, the Tsen rank of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.

Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students.

From Reviews of the German version:

This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the

framework of the development of carefully selected problems.

- Stefan Porubsky, Mathematical Reviews

---

The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.

The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.

The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, the Tsen rank of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.

Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students.

 

From Reviews of the German version:

This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the

framework of the development of carefully selected problems.

- Stefan Porubsky, Mathematical Reviews

---

The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.

The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.

The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, the Tsen rank of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.

Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students.

 

From Reviews of the German version:

This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the

framework of the development of carefully selected problems.

- Stefan Porubsky, Mathematical Reviews

Content:
Front Matter....Pages i-ix
Ordered Fields and Real Fields....Pages 1-14
Hilbert's Seventeenth Problem and the Real Nullstellensatz....Pages 15-27
Orders and Quadratic Forms....Pages 29-38
Absolute Values on Fields....Pages 39-63
Residue Class Degree and Ramification Index....Pages 65-74
Local Fields....Pages 75-91
Witt Vectors....Pages 93-108
The Tsen Rank of a Field....Pages 109-126
Fundamentals of Modules....Pages 127-149
Wedderburn Theory....Pages 151-181
Crossed Products....Pages 183-222
The Brauer Group of a Local Field....Pages 223-238
Local Class Field Theory....Pages 239-251
Semisimple Representations of Finite Groups....Pages 253-281
The Schur Group of a Field....Pages 283-298
Back Matter....Pages 299-336

---

The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.

The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.

The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, the Tsen rank of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.

Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students.

 

From Reviews of the German version:

This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the

framework of the development of carefully selected problems.

- Stefan Porubsky, Mathematical Reviews

Content:
Front Matter....Pages i-ix
Ordered Fields and Real Fields....Pages 1-14
Hilbert's Seventeenth Problem and the Real Nullstellensatz....Pages 15-27
Orders and Quadratic Forms....Pages 29-38
Absolute Values on Fields....Pages 39-63
Residue Class Degree and Ramification Index....Pages 65-74
Local Fields....Pages 75-91
Witt Vectors....Pages 93-108
The Tsen Rank of a Field....Pages 109-126
Fundamentals of Modules....Pages 127-149
Wedderburn Theory....Pages 151-181
Crossed Products....Pages 183-222
The Brauer Group of a Local Field....Pages 223-238
Local Class Field Theory....Pages 239-251
Semisimple Representations of Finite Groups....Pages 253-281
The Schur Group of a Field....Pages 283-298
Back Matter....Pages 299-336
....