## Algebraic Number TheoryFrom the reviews: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994"... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995 |

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### Sommario

7 | |

Number Fields | 19 |

3 Dedekind Rings | 28 |

4 Valuations | 45 |

5 Harmonic Analysis on Local and Global Fields | 63 |

6 Hecke LSeries and the Distribution of Prime Ideals | 70 |

Class Field Theory | 90 |

2 Complex Multiplication | 107 |

2 Galois Cohomology of Local and Global Fields | 168 |

Abelian Fields | 192 |

3 Iwasawas Theory of IExtensions | 206 |

4 padic LFunctions | 212 |

Artin LFunctions and Galois Module Structure | 219 |

2 Galois Module Structure and Artin Root Numbers | 234 |

Fields Domains and Complexes | 237 |

Tables | 245 |

5 Simple Algebras | 131 |

6 Explicit Reciprocity Laws and Symbols | 137 |

7 Further Results of Class Field Theory | 145 |

Cohomology of Profinite Groups | 151 |

251 | |

Author Index 263 | 262 |

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### Parole e frasi comuni

abelian extension algebraic number field arbitrary Artin basis called Chap character class field theory class group class number closed complex compute conductor conjecture consider contains continuous corresponding cyclic cyclic extension cyclotomic defined definition denotes determined direct discriminant divisor elements embedding equal equation exact Example exists factor finite finite extension fixed formula function Furthermore G-module Galois group given global group G Hence Hilbert homomorphism implies induces infinite integers irreducible isomorphism L-functions Let G Let L/K Main reference maximal means module morphism multiplication natural number norm normal extension p-adic places polynomial prime ideal principal problem profinite group Proof properties Proposition proved quadratic ramified representation residue respect restriction ring roots of unity sequence shows structure subgroup symbol Theorem trivial unique unit unramified valuation values