Algebraic Number Theory

1. Front Matter

   Pages i-xiii

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2. Basic Theory

   1. Front Matter

      Pages 1-1

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   2. Algebraic Integers

      • Serge Lang

      Pages 3-29

   3. Completions

      • Serge Lang

      Pages 31-55

   4. The Different and Discriminant

      • Serge Lang

      Pages 57-69

   5. Cyclotomic Fields

      • Serge Lang

      Pages 71-98

   6. Parallelotopes

      • Serge Lang

      Pages 99-122

   7. The Ideal Function

      • Serge Lang

      Pages 123-135

   8. Ideles and Adeles

      • Serge Lang

      Pages 137-154

   9. Elementary Properties of the Zeta Function and L-series

      • Serge Lang

      Pages 155-170

3. Class Field Theory

   1. Front Matter

      Pages 171-177

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   2. Norm Index Computations

      • Serge Lang

      Pages 179-195

   3. The Artin Symbol, Reciprocity Law, and Class Field Theory

      • Serge Lang

      Pages 197-212

   4. The Existence Theorem and Local Class Field Theory

      • Serge Lang

      Pages 213-227

   5. L-Series Again

      • Serge Lang

      Pages 229-239

4. Analytic Theory

   1. Front Matter

      Pages 241-243

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   2. Functional Equation of the Zeta Function, Hecke’s Proof

      • Serge Lang

      Pages 245-273

   3. Functional Equation, Tate’s Thesis

      • Serge Lang

      Pages 275-301

   4. Density of Primes and Tauberian Theorem

      • Serge Lang

      Pages 303-319

   5. The Brauer-Siegel Theorem

      • Serge Lang

      Pages 321-330

The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e. g. the class field theory on which I make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collec­ tion of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of Weber, Hasse, Hecke, and Hilbert's Zahlbericht. It seems that over the years, everything that has been done has proved useful, theo­ retically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more. The point of view taken here is principally global, and we deal with local fields only incidentally. For a more complete treatment of these, cf. Serre's book Corps Locaux. There is much to be said for a direct global approach to number fields. Stylistically, I have intermingled the ideal and idelic approaches without prejudice for either. I also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods).

• algebraic number theory
• analytic number theory
• number theory
• zeta function

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

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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