## Algebraic Number TheoryThe 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). |

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### Contents

3 | |

11 | |

Dedekind rings | 18 |

Explicit factorization of a prime | 27 |

Polynomials in complete fields | 41 |

Unramified extensions | 48 |

CHAPTER III | 57 |

The discriminant | 64 |

The global cyclic norm index | 192 |

Existence of a conductor for the Artin symbo | 200 |

Class fields | 206 |

CHAPTER XI | 213 |

Local class field theory and the ramification theorem | 219 |

Infinite divisibility of the universal norms | 225 |

Artin nonabelian Lseries | 232 |

CHAPTER XIII | 243 |

CHAPTER IV | 71 |

CHAPTER V | 99 |

Lattice points in parallelotopes | 110 |

A volume computation | 116 |

31 | 117 |

The Ideal Function | 123 |

The number of ideals in a given class | 129 |

CHAPTER VII | 137 |

Generalized ideal class groups relations with idele classes | 145 |

in the idele classes | 151 |

Zeta function of a number field | 159 |

45 | 164 |

Density of primes in arithmetic progressions | 166 |

CHAPTER IX | 179 |

Exponential and logarithm functions | 185 |

A special computation | 250 |

Application to the BrauerSiegel theorem | 260 |

Functional Equation Tates Thesis | 275 |

Local computations | 282 |

Global additive duality and RiemannRoch theorem | 289 |

Global computations | 297 |

Tauberian theorem for Dirichlet series | 310 |

CHAPTER XVI | 321 |

End of the proofs | 327 |

57 | 329 |

Explicit Formulas | 331 |

Bibliography | 340 |

351 | |

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### Common terms and phrases

a e k abelian extension adele apply archimedean absolute values Artin map assertion assume automorphism bounded Chapter character class field theory coefficients compact complex numbers compute concludes the proof contained converges Corollary coset cyclic cyclotomic decomposition group define denote Dirichlet series discriminant embedded equal factor group finite extension finite number follows formula fractional ideal functional equation fundamental domain Galois extension Galois group given group G Hence homomorphism ideal class group idele induced inequality integral closure isomorphism kernel L-series lattice Lemma Let f Let G Let K/k maximal ideal Mk-divisor mod q multiplicative group n-th roots non-zero norm index notation number field number of elements obtain open subgroup p-adic polynomial positive integer prime ideal prime number Proposition quasi-character Re(s reciprocity law roots of unity satisfying splits completely subset surjective trivial unit unramified variables vector whence write zeta function