## Algebraic Number TheoryThis book originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists. |

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### Contents

Algebraic Foundations | 8 |

I2 Integrality and Noetherian properties | 26 |

Dedekind Domains | 35 |

II2 Valuations and absolute values | 58 |

II3 Completions | 70 |

II4 Module theory over a Dedekind domain | 87 |

Extensions | 102 |

III2 Discriminants and differents | 120 |

VI3 Quadratic fields revisited | 220 |

VI4 Gauss sums | 231 |

VI5 Elliptic curves | 241 |

Diophantine Equations | 251 |

VII2 Quadratic forms | 254 |

VII3 Cubic equations | 269 |

Lfunctions | 277 |

VIII2 The Dedekind zeta function | 283 |

III3 Nonramified and tamely ramified extensions | 132 |

III4 Ramification in Galois extensions | 142 |

Classgroups and Units | 152 |

IV2 Lattices in Euclidean space | 156 |

IV3 Classgroups | 164 |

IV4 Units | 168 |

Fields of low degree | 175 |

V2 Biquadratic fields | 193 |

V3 Cubic and sextic fields | 198 |

Cyclotomic Fields | 205 |

VI2 Characters | 213 |

VIII3 Dirichlet Lfunctions | 295 |

VIII4 Primes in an arithmetic progression | 297 |

VIII5 Evaluation of L1𝜒 and explicit class number formulae for cyclotomic fields | 299 |

VIII6 Quadratic fields yet again | 306 |

VIII7 Brauer relations | 309 |

Characters of Finite Abelian Groups | 327 |

Exercises | 335 |

349 | |

Glossary of Theorems | 352 |

353 | |

### Common terms and phrases

algebraic integers algebraic number field arithmetic automorphism basis Cauchy sequence Chapter class number classgroup clearly complete conclude consider converges coprime Corollary cyclotomic field decomposition Dedekind domain deduce define denote a finite discrete absolute value discriminant distinct prime element equation equivalent factorisation field of fractions finite commutative finite separable extension fractional ideal fractional o-ideal free o-module fundamental unit Galois extension Galois group given hence homomorphism idempotents imaginary implies induces integral closure inverse irreducible isomorphism let p denote maximal module monic moreover Noetherian ring non-ramified non-trivial non-zero norm notation number theory obtain odd prime polynomial positive integer prime ideal prime number primitive principal ideal domain Proof prove quadratic fields quadratic forms real number residue class character resp ring of algebraic root of unity separable extension subfield subgroup suppose surjective valuation valued fields write