## Algebraic Number TheoryCareful organization and clear, detailed proofs make this book ideal either for classroom use or as a stimulating series of exercises for mathematically-minded individuals. Modern abstract techniques focus on introducing elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields. |

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### Contents

Elementary Valuation Theory | 1 |

Extension of Valuations | 41 |

Local Fields | 72 |

Ordinary Arithmetic Fields | 118 |

Global Fields | 185 |

Quadratic Fields | 233 |

Cyclotomic Fields | 255 |

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a e F algebraic closure algebraic number ﬁeld archimedean prime class number closure compact completes the proof composition maps Consider Corollary Dedekind ring deﬁned deﬁnition denote diﬁerent discrete prime divisor discriminant divisor of F Exercise exists exponential valuation extension E/F extension of F fact factorization ﬁeld F ﬁnd ﬁnite extension ﬁrst ﬁxed follows Furthermore Galois extension global ﬁeld homomorphism ideal 91 ideal of F inﬁnite integral basis integral ideal integrally closed irreducible isomorphism lattice lemma Let F lsee monic Moreover multiplicative nonarchimedean prime divisor nontrivial norm notation prime divisor prime element prime ideal principal ideal domain product formula product topology Proposition quadratic ﬁeld quotient ﬁeld residue class ﬁeld restricted direct product roots of unity satisﬁes subﬁeld subgroup Suppose that F tamely ramiﬁed theory topology totally ramiﬁed trivial unique extension unramiﬁed extension valuation of F vp(a