## Algebraic number theory |

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### Contents

Algebraic background | 9 |

Algebraic numbers | 38 |

Quadratic and cyclotomic fields | 59 |

Copyright | |

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### Common terms and phrases

additive group algebraic integer algebraic number calculate chapter class-group class-number coefficients complex numbers compute conjugates contradiction coprime Corollary criterion cyclic cyclotomic fields deduce define denote discriminant divides equation equivalent example exists factoriz factorization into irreducibles factorization of elements Fermat's Last Theorem finite number follows fractional ideals free abelian group fundamental domain Gauss Hence highest common factor homomorphism implies integral basis integral domain isomorphic Kummer lattice Lemma maximal minimum polynomial Minkowski's theorem monic monomorphisms multiple noetherian nomial notation number field number theory odd prime poly prime factorization prime ideals principal ideal proper factors properties Proposition prove quadratic fields quadratic residue modulo quotient rational integer rational prime result ring of integers roots of unity satisfies solutions squarefree subgroup subring subset Suppose symmetric polynomials taking norms U(Zp unique factorization uniqueness of factorization unit vector space Z-basis zero