## A Brief Guide to Algebraic Number TheoryThis account of Algebraic Number Theory is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included. |

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### Contents

Valuations | 31 |

Special fields | 55 |

Analytic methods | 79 |

Class Field Theory | 98 |

Appendix | 117 |

Exercises | 135 |

143 | |

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### Common terms and phrases

abelian extension abelian group adeles algebraic number field Archimedean valuation argument assertion assume base character choose class field theory class number coefficients complex conductor conjugates contains coprime Corollary to Theorem corresponding coset cyclic cyclotomic field Dedekind define denote Dirichlet Dirichlet density divides elements equivalent extension K/k finite set fixed field fractional ideal Gal(K/k Galois group gives Haar measure hence Hilbert symbol homomorphism ideal class group ideles implies induced infinite places integral ideal isomorphism kernel Lemma 18 m-th root minimal monic polynomial monic polynomial multiplication Noetherian non-trivial non-zero Norm normal Normp notation o-submodule obtain prime ideals principal ideal Proof Let properties prove quadratic form quotient ramified residue classes result right hand side ring of integers roots of unity satisfies span splits subgroup Suppose Theorem 27 topology trivial unique unit unramified Vk/k write Z-module