Algebraic Number Theory

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 1 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 W eber, 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, 1 have intermingled the ideal and idelic approaches without prejudice for either. 1 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).

One General Basic Theory. - I Algebraic Integers. - II Completions. - III The Different and Discriminant. - IV Cyclotomic Fields. - V Parallelotopes. - VI The Ideal Function. - VII Ideles and Adeles. - VIII Elementary Properties of the Zeta Function and L-series. - Two Class Field Theory. - IX Norm Index Computations. - X The Artin Symbol, Reciprocity Law, and Class Field Theory. - XI The Existence Theorem and Local Class Field Theory. - XII L-series Again. - Three Analytic Theory. - XIII Functional Equation of the Zeta Function, Hecke s Proof. - XIV Functional Equation, Tate s Thesis. - XV Density of Primes and Tauberian Theorem. - XVI The Brauer-Siegel Theorem. - XVII Explicit Formulas.