Algebraic Number Theory

Serge Lang (auth.)9780387942254, 0-387-94225-4, 3540942254

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. 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. This new edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

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Table of contents :
Front Matter….Pages i-xiii
Front Matter….Pages 1-1
Algebraic Integers….Pages 3-29
Completions….Pages 31-55
The Different and Discriminant….Pages 57-69
Cyclotomic Fields….Pages 71-98
Parallelotopes….Pages 99-122
The Ideal Function….Pages 123-135
Ideles and Adeles….Pages 137-154
Elementary Properties of the Zeta Function and L -series….Pages 155-170
Front Matter….Pages 171-177
Norm Index Computations….Pages 179-195
The Artin Symbol, Reciprocity Law, and Class Field Theory….Pages 197-212
The Existence Theorem and Local Class Field Theory….Pages 213-227
L -Series Again….Pages 229-239
Front Matter….Pages 241-243
Functional Equation of the Zeta Function, Hecke’s Proof….Pages 245-273
Functional Equation, Tate’s Thesis….Pages 275-301
Density of Primes and Tauberian Theorem….Pages 303-319
The Brauer-Siegel Theorem….Pages 321-330
Explicit Formulas….Pages 331-349
Back Matter….Pages 351-354