## The Theory of Algebraic Number FieldsConstance Reid, in Chapter VII of her book Hilbert, tells the story of the writing of the Zahlbericht, as his report entitled Die Theorie der algebra is chen Zahlkorper has always been known. At its annual meeting in 1893 the Deutsche Mathematiker-Vereinigung (the German Mathematical Society) invited Hilbert and Minkowski to prepare a report on the current state of affairs in the theory of numbers, to be completed in two years. The two mathematicians agreed that Minkowski should write about rational number theory and Hilbert about algebraic number theory. Although Hilbert had almost completed his share of the report by the beginning of 1896 Minkowski had made much less progress and it was agreed that he should withdraw from his part of the project. Shortly afterwards Hilbert finished writing his report on algebraic number fields and the manuscript, carefully copied by his wife, was sent to the printers. The proofs were read by Minkowski, aided in part by Hurwitz, slowly and carefully, with close attention to the mathematical exposition as well as to the type-setting; at Minkowski's insistence Hilbert included a note of thanks to his wife. As Constance Reid writes, "The report on algebraic number fields exceeded in every way the expectation of the members of the Mathemati cal Society. They had asked for a summary of the current state of affairs in the theory. They received a masterpiece, which simply and clearly fitted all the difficult developments of recent times into an elegantly integrated theory. |

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### Contents

VIII | 3 |

IX | 4 |

X | 5 |

XI | 9 |

XII | 11 |

XIII | 14 |

XIV | 17 |

XV | 20 |

XCII | 161 |

XCIII | 162 |

XCIV | 163 |

XCV | 167 |

XCVI | 168 |

XCVIII | 169 |

XCIX | 171 |

C | 175 |

XVI | 22 |

XVII | 25 |

XVIII | 28 |

XIX | 30 |

XX | 31 |

XXI | 33 |

XXII | 38 |

XXIII | 41 |

XXIV | 43 |

XXV | 45 |

XXVI | 49 |

XXVII | 51 |

XXVIII | 53 |

XXIX | 56 |

XXXI | 60 |

XXXII | 62 |

XXXIV | 65 |

XXXV | 67 |

XXXVI | 69 |

XXXVII | 72 |

XXXVIII | 73 |

XXXIX | 74 |

XL | 79 |

XLI | 81 |

XLIII | 82 |

XLIV | 83 |

XLV | 84 |

XLVI | 85 |

XLVII | 86 |

XLIX | 87 |

L | 89 |

LI | 90 |

LII | 93 |

LIII | 94 |

LV | 97 |

LVI | 98 |

LVII | 101 |

LVIII | 105 |

LIX | 106 |

LX | 108 |

LXI | 109 |

LXII | 111 |

LXIII | 115 |

LXIV | 116 |

LXV | 118 |

LXVI | 119 |

LXVIII | 121 |

LXIX | 125 |

LXX | 126 |

LXXI | 127 |

LXXIII | 128 |

LXXIV | 131 |

LXXV | 133 |

LXXVI | 135 |

LXXVII | 136 |

LXXX | 138 |

LXXXI | 139 |

LXXXIII | 140 |

LXXXIV | 142 |

LXXXV | 144 |

LXXXVI | 146 |

LXXXVII | 147 |

LXXXVIII | 149 |

LXXXIX | 152 |

XC | 155 |

XCI | 157 |

CI | 176 |

CII | 177 |

CIII | 178 |

CIV | 181 |

CV | 184 |

CVI | 187 |

CVII | 188 |

CIX | 193 |

CX | 194 |

CXI | 195 |

CXIII | 199 |

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CXVI | 208 |

CXVII | 211 |

CXVIII | 213 |

CXIX | 215 |

CXX | 217 |

CXXI | 219 |

CXXII | 220 |

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CXXV | 229 |

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CXXVII | 233 |

CXXVIII | 243 |

CXXIX | 248 |

CXXX | 253 |

CXXXI | 257 |

CXXXII | 259 |

CXXXIII | 262 |

CXXXIV | 264 |

CXXXV | 265 |

CXXXVI | 266 |

CXXXVII | 269 |

CXXXVIII | 270 |

CXL | 271 |

CXLI | 273 |

CXLII | 280 |

CXLIII | 282 |

CXLIV | 285 |

CXLV | 286 |

CXLVI | 289 |

CXLVII | 290 |

CXLVIII | 293 |

CXLIX | 296 |

CL | 298 |

CLI | 300 |

CLII | 301 |

CLIII | 303 |

CLIV | 305 |

CLV | 306 |

CLVI | 308 |

CLVII | 309 |

CLVIII | 313 |

CLIX | 315 |

CLX | 318 |

CLXI | 321 |

CLXII | 324 |

CLXIII | 325 |

CLXIV | 327 |

CLXV | 332 |

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### Common terms and phrases

abelian field according to Theorem algebraic integers algebraic number algebraic number fields ambig classes automorphisms character set class number congruence congruent modulo conjugate cyclic extension cyclic field decomposition group Dedekind deduce denote determined Dirichlet equation exponent factorisation fc(C field fc field fc(M field of degree follows from Theorem formula Galois number field genera greatest common divisor hence Hilbert ideal of degree inertia field inertia group integer polynomial Kronecker l-th power l-th roots Lemma Math mod I2 norm residue number h number theory obtain odd prime number order ideal primary number prime factor prime ideal principal genus principal ideal proof of Theorem prove quadratic field ramification group rational integer rational number rational prime number reciprocity law regular cyclotomic field Reine Angew relative discriminant relative norm residue modulo respectively root number roots of unity Sect subfield symbol Theorem 151 Uber unit bundle Werke Wiss Zahlbericht Zahlen

### References to this book

The Legacy of Mario Pieri in Geometry and Arithmetic Elena Anne Marchisotto,James T. Smith Limited preview - 2007 |

Computational Optimization: A Tribute to Olvi Mangasarian, Volume 1 Olvi L. Mangasarian,Jong-Shi Pang No preview available - 1999 |