A formula for the discriminant of number fields

Pei-Chu Hu, Zhuan Ye
2010 Proceedings of the American Mathematical Society  
We obtain a formula for the discriminant D κ/Q of an algebraic number field κ in terms of a ratio of the first two coefficients of the Taylor series of ζ κ at 1/2. Let κ be a number field of degree n = r 1 + 2r 2 , where r 1 , r 2 are the number of real, complex places respectively. The Dedekind ζ-function of the number field κ is defined by the series where a varies over the non-zero integral ideals of κ and N (a) denotes the absolute norm of a. Denote by D κ/Q the discriminant of κ. The
more » ... ant of κ. The Dedekind function ζ κ (s) admits a holomorphic continuation with the exclusion of a simple pole at s = 1 and satisfies the following functional equation (1) ζ κ (1 − s) = A(s)ζ κ (s), where A(s) = |D κ/Q | s− 1 2 cos πs 2 r 1 +r 2 sin πs 2 r 2 2 (1−s)n π −sn Γ n (s). A straightforward computation gives A(1/2) = 1. Furthermore, let β 1,κ ( = 0), γ 1,κ be defined by the Taylor expansion of ζ κ (s) at s = 1/2, i.e., (2) ζ κ (s) = β 1,κ s − 1 2 μ
doi:10.1090/s0002-9939-2010-10546-4 fatcat:jasu7xuxxzftbkuk2xhepeuhny