# Notations and Conventions Concerning Quadratic Number Fields

• Gerard van der Geer
Chapter
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
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