## Class Field TheoryThis classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory ... In this revised edition, two mathematical additions complementing the exposition of the original text are made. The new edition also contains several new footnotes, additional references, and historical comments. |

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### Contents

The First Fundamental Inequality | 11 |

Second Fundamental Inequality | 19 |

Reciprocity Law | 35 |

The Existence Theorem | 55 |

Connected Component of Idele Classes | 65 |

The GrunwaldWang Theorem | 73 |

Higher Ramification Theory | 83 |

Explicit Reciprocity Laws | 109 |

Group Extensions | 127 |

Abstract Class Field Theory | 143 |

Weil Groups | 167 |

191 | |

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2-cocycle a e G abelian extension algebraic archimedean primes automorphism Axiom Brauer group Chapter character class field theory class formation cocycle class cohomology groups commutative compact concludes the proof consequently contains COROLLARY corresponding cyclic extension defined definition denote extension of degree factor group finite extension finite index function fields fundamental class Galois group GK/F global field group extension group G Hence homomorphism idele class induced integer invariant invp irreducible isomorphism kernel Lemma Let G Let K/k module multiplicative group n-th power n-th roots neighborhood norm residue symbol norm subgroup normal extension normal layer K/F number fields obtain open subgroup ordx prime degree properties PROPOSITION prove reciprocity law residue class field root of unity second inequality set of primes shows subfield subgroup of finite subgroup of G trivial unramified W-diagram Weil group