Notations and Conventions Concerning Quadratic Number Fields

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