Algebraic Number Theory

• Serge Lang
Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 110)

1. Front Matter
Pages i-xiii
2. Basic Theory

1. Front Matter
Pages 1-1
2. Serge Lang
Pages 3-29
3. Serge Lang
Pages 31-55
4. Serge Lang
Pages 57-69
5. Serge Lang
Pages 71-98
6. Serge Lang
Pages 99-122
7. Serge Lang
Pages 123-135
8. Serge Lang
Pages 137-154
9. Serge Lang
Pages 155-170
3. Class Field Theory

1. Front Matter
Pages 171-177
2. Serge Lang
Pages 179-195
3. Serge Lang
Pages 197-212
4. Serge Lang
Pages 213-227
5. Serge Lang
Pages 229-239
4. Analytic Theory

1. Front Matter
Pages 241-243
2. Serge Lang
Pages 245-273
3. Serge Lang
Pages 275-301
4. Serge Lang
Pages 303-319
5. Serge Lang
Pages 321-330
6. Serge Lang
Pages 331-349
5. Back Matter
Pages 351-354

Introduction

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).

Keywords

algebraic number theory analytic number theory number theory zeta function

Authors and affiliations

• Serge Lang
• 1
1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-0296-4
• Copyright Information Springer-Verlag New York 1986
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4684-0298-8
• Online ISBN 978-1-4684-0296-4
• Series Print ISSN 0072-5285
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