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### Synopses & Reviews

#### Publisher Comments

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

#### Review

Second Edition S. Lang Algebraic Number Theory "This book is the second edition of Lang's famous and indispensable book on algebraic number theory. The major change from the previous edition is that the last chapter on explicit formulas has been completely rewritten. In addition, a few new sections have been added to the other chapters . . . Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception."--MATHEMATICAL REVIEWS

#### Review

Second Edition

*S. Lang*

*Algebraic Number Theory*

*"This book is the second edition of Lang's famous and indispensable book on algebraic number theory. The major change from the previous edition is that the last chapter on explicit formulas has been completely rewritten. In addition, a few new sections have been added to the other chapters . . . Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception."--*MATHEMATICAL REVIEWS

#### Synopsis

This is a second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. "Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception."---MATHEMATICAL REVIEWS

### Table of Contents

**Part I: General Basic Theory:**Algebraic Integers. Completions. The Different and Discriminant. Cyclotomic Fields. Paralellotopes. The Ideal Function. Ideles and Adeles. Elementary Properties of the Zeta Function and

*L*-series.-

**Part II: Class Field Theory:**Norm Index Computations. The Artin Symbol, Reciprocity Law, and Class Field Theory. The Existence Theorem and Local Class Field Theory.

*L*-series Again.-

**Part III: Analytic**

**Theory:**Functional Equation of the Zeta Function, Hecke's Proof. Functional Equation, Tate's Thesis. Density of Primes and Tauberian Theorem. The Brauer-Siegel Theorem. Explicit Formulas.- Bibliography.- Index.

### Product Details

- ISBN:
- 9780387942254
- Binding:
- Hardcover
- Publication date:
- 07/19/2000
- Publisher:
- Springer
- Series info:
- Applied Mathematical Sciences
- Language:
- English
- Edition:
- 2
- Pages:
- 357
- Height:
- .98IN
- Width:
- 6.42IN
- LCCN:
- 93050625
- Series:
- Applied Mathematical Sciences
- Series Number:
- 110
- Number of Units:
- 1
- Illustration:
- Yes
- Copyright Year:
- 1994
- Series Volume:
- 110
- UPC Code:
- 2800387942256
- Author:
- Media Run Time:
- B
- Subject:
- Algebraic number theory
- Subject:
- mathematics and statistics
- Subject:
- Mathematics
- Subject:
- Language, literature and biography