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Notations and Conventions Concerning Quadratic Number Fields

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

Abstract

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:
$$ D = \prod\limits_{p|D} {D(p)} $$
.

Keywords

Number Field Quadratic Number Real Number Field Real Quadratic Field Fundamental Discriminant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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