## Lectures on the Theory of Algebraic Numbers. . . if one wants to make progress in mathematics one should study the masters not the pupils. N. H. Abel Heeke was certainly one of the masters, and in fact, the study of Heeke L series and Heeke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book, and Heeke's Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "To improve upon Heeke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. " We have tried to remain as close as possible to the original text in pre serving Heeke's rich, informal style of exposition. In a very few instances we have substituted modern terminology for Heeke's, e. g. , "torsion free group" for "pure group. " One problem for a student is the lack of exercises in the book. However, given the large number of texts available in algebraic number theory, this is not a serious drawback. In particular we recommend Number Fields by D. A. Marcus (Springer-Verlag) as a particularly rich source. We would like to thank James M. Vaughn Jr. and the Vaughn Foundation Fund for their encouragement and generous support of Jay R. Goldman without which this translation would never have appeared. Minneapolis George U. Brauer July 1981 Jay R. |

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### Contents

I | 1 |

II | 6 |

III | 10 |

IV | 13 |

V | 16 |

VI | 20 |

VII | 22 |

VIII | 24 |

XXXIV | 108 |

XXXV | 113 |

XXXVI | 116 |

XXXVII | 122 |

XXXVIII | 125 |

XXXIX | 132 |

XL | 139 |

XLI | 143 |

IX | 28 |

X | 30 |

XI | 34 |

XII | 40 |

XIII | 42 |

XIV | 45 |

XV | 48 |

XVI | 50 |

XVII | 54 |

XVIII | 57 |

XIX | 59 |

XX | 63 |

XXI | 68 |

XXII | 71 |

XXIII | 73 |

XXIV | 77 |

XXV | 83 |

XXVI | 85 |

XXVII | 87 |

XXVIII | 91 |

XXIX | 94 |

XXX | 98 |

XXXI | 100 |

XXXII | 102 |

XXXIII | 105 |

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### Common terms and phrases

algebraic integers algebraic number field basis elements basis number belongs called class number complete system congruence Consequently Conversely coprime cosets decomposition laws defined definition denominator denotes divides divisible equal equation equivalent exactly exponents factor group finite follows form a subgroup formula function fundamental theorem Gauss sums greatest common divisor hence holds ideal classes independent infinite Abelian group integral ideal integral polynomial irreducible Lemma linear modulo Moreover number complexes number theory obtain obviously odd ideal odd prime positive number positive primes prime factors prime ideal principal ideal product of powers proof proved quadratic field Quadratic Reciprocity Law quadratic residue mod quotient rational integers rational numbers rational prime relative degree relative norm relatively prime represented residue classes mod respect roots of unity runs singular number singular primary numbers solutions solvable square symbol system of residues theta-series totally positive uniquely determined unit element variables zero