## Algebraic Number Theory, Volume 62From the reviews of the first printing, published as Volume 62 of the Encyclopaedia of Mathematical Sciences: "... 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 "... Koch's book is written mostly for non-specialists. It is an up-to-date account of the subject dealing with mostly general questions. Special results appear only as illustrating examples for the general features of the theory. It is supposed that the reader has good general background in the fields of modern (abstract) algebra and elementary number theory. We recommend this volume mainly to graduate studens and research mathematicians." Acta Scientiarum Mathematicarum, 1993 |

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### Sommario

Basic Number Theory | 8 |

1 Orders in Algebraic Number Fields | 9 |

2 Rings with Divisor Theory | 22 |

3 Dedekind Rings | 27 |

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 |

3 Extensions with Given Galois Groups | 182 |

Abelian Fields | 192 |

1 The Integers of an Abelian Field | 193 |

2 The Arithmetical Class Number Formula | 195 |

3 Iwasawas Theory of ΓExtensions | 206 |

4 padic LFunctions | 211 |

Artin LFunctions and Galois Module Structure | 219 |

1 Artin LFunctions | 222 |

1 The Main Theorems of Class Field Theory | 92 |

2 Complex Multiplication | 107 |

3 Cohomology of Groups | 112 |

4 Proof of the Main Theorems of Class Field Theory | 121 |

5 Simple Algebras | 131 |

6 Explicit Reciprocity Laws and Symbols | 137 |

7 Further Results of Class Field Theory | 145 |

Galois Groups | 150 |

1 Cohomology of Profinite Groups | 151 |

2 Galois Cohomology of Local and Global Fields | 168 |

2 Galois Module Structure and Artin Root Numbers | 234 |

Fields Domains and Complexes | 237 |

Quadratic Residues | 240 |

Locally Compact Groups | 241 |

Bernoulli Numbers | 243 |

Tables | 245 |

251 | |

263 | |

266 | |

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### Parole e frasi comuni

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