Notations and Conventions Concerning Quadratic Number Fields

  • Gerard van der Geer
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 16)


An integer D is called a fundamental discriminant if it is the discriminant of (the maximal order of) a quadratic number field. For a fundamental discriminant we denote by χ D the primitive Dirichlet character mod ∣D∣ satisfying \( {\chi_D}(p) = \left( {\frac{D}{p}} \right) \) for any odd prime, χ D (2) = +1 (resp. - 1) if D ≡ 1 (mod 8) (resp. D ≡ 5 (mod 8)) and χ D (- 1) = sgn D. A fundamental discriminant is called a prime discriminant if it is divisible by only one prime (i.e. D = - 4, + 8, - 8, p with p a prime ≡ 1 (mod 4), -p with p a prime ≡ 3 (mod 4)). Every fundamental discriminant is a product of prime discriminants:
$$ D = \prod\limits_{p|D} {D(p)} $$


Number Field Quadratic Number Real Number Field Real Quadratic Field Fundamental Discriminant 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Gerard van der Geer
    • 1
  1. 1.Mathematisch InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands

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