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Quadratic and cyclotomic fields
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additive group algebraic integer algebraic number calculate chapter class-group class-number coefficients complex numbers compute conjugates consider contradiction coprime Corollary criterion cyclic cyclotomic fields deduce define 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's lattice Lemma maximal minimum polynomial Minkowski's theorem monic monomorphisms multiplication noetherian nomial non-zero 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 vector space Z-basis zero