## Algebraic number theoryIdeal either for classroom use or as exercises for mathematically-minded individuals, this text introduces 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 |

Copyright | |

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a e F algebraic closure algebraic number ﬁeld archimedean prime basis for E/F class number closure completes the proof composition maps Consider Corollary deﬁned deﬁnition denote discrete prime divisor discriminant divisor of F 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 21 ideal class ideal of F inﬁnite integral basis integral ideal integrally closed irreducible isomorphism lattice lemma Let F monic Moreover Newton polygon nonarchimedean prime divisor nontrivial norm notation OE ideal P-complete 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 topology totally ramiﬁed trivial unique extension unramiﬁed extension valuation of F vp(a