## Algebraic Number TheoryThe present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e. g. the class field theory on which 1 make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collec tion of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of W eber, Hasse, Hecke, and Hilbert's Zahlbericht. It seems that over the years, everything that has been done has proved useful, theo retically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more. The point of view taken here is principally global, and we deal with local fields only incidentally. For a more complete treatment of these, cf. Serre's book Corps Locaux. There is much to be said for a direct global approach to number fields. Stylistically, 1 have intermingled the ideal and idelic approaches without prejudice for either. 1 also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods). |

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### Contents

II | 3 |

III | 4 |

IV | 8 |

V | 11 |

VI | 12 |

VII | 18 |

VIII | 22 |

IX | 27 |

XLIX | 200 |

L | 206 |

LI | 213 |

LII | 215 |

LIII | 217 |

LIV | 219 |

LV | 224 |

LVI | 226 |

X | 29 |

XI | 31 |

XII | 41 |

XIII | 45 |

XIV | 48 |

XV | 51 |

XVI | 57 |

XVII | 62 |

XVIII | 64 |

XIX | 71 |

XX | 76 |

XXI | 82 |

XXII | 96 |

XXIII | 99 |

XXIV | 110 |

XXV | 116 |

XXVI | 119 |

XXVII | 123 |

XXVIII | 128 |

XXIX | 129 |

XXX | 137 |

XXXI | 139 |

XXXII | 140 |

XXXIII | 145 |

XXXIV | 151 |

XXXV | 152 |

XXXVI | 155 |

XXXVII | 159 |

XXXVIII | 162 |

XXXIX | 166 |

XL | 170 |

XLI | 173 |

XLII | 179 |

XLIII | 185 |

XLIV | 187 |

XLV | 190 |

XLVI | 193 |

XLVII | 195 |

XLVIII | 197 |

LVII | 229 |

LVIII | 232 |

LIX | 236 |

LX | 241 |

LXI | 245 |

LXII | 250 |

LXIII | 253 |

LXIV | 260 |

LXV | 262 |

LXVI | 273 |

LXVII | 275 |

LXVIII | 276 |

LXIX | 278 |

LXX | 280 |

LXXI | 282 |

LXXII | 287 |

LXXIII | 289 |

LXXIV | 292 |

LXXV | 297 |

LXXVI | 303 |

LXXVII | 304 |

LXXVIII | 310 |

LXXIX | 312 |

LXXX | 315 |

LXXXI | 321 |

LXXXII | 322 |

LXXXIII | 323 |

LXXXIV | 325 |

LXXXV | 327 |

LXXXVI | 328 |

LXXXVII | 331 |

LXXXVIII | 333 |

LXXXIX | 337 |

XC | 344 |

XCI | 348 |

353 | |

355 | |

### Common terms and phrases

abelian extension adele algebraic integers archimedean absolute values Artin map assertion assume automorphism bounded Chapter character class field theory coefficients compact complex numbers concludes the proof conjugate contained converges Corollary cyclic cyclotomic decomposition group Dedekind ring define degree denote Dirichlet series discriminant embedded equal exists an element factor group finite extension finite number follows formula Fourier fractional ideal functional equation fundamental domain Galois extension Galois group given group G Hence homomorphism ideal class group idele induced inequality integral closure isomorphism kernel L-series lattice Lemma Let f Let K/k maximal ideal module multiplicative group non-zero notation number field number of elements obtain open subgroup p-adic positive integer prime ideal prime number Proposition prove quasi-character quotient field Re(s residue class field roots of unity satisfies splits completely subset surjective Theorem trivial unit unramified variables vector whence write zeta function