The 4-class ranks of quadratic fields

Abstract

Let K be a quadratic extension field of the rational numbers Q. Let C K be the 2-class group of K in the narrow sense. It is a classical result that rank C K= t l , where t is the number of primes that ramify in K/Q. Now let C~ = {ai: a e CK}, and let R K denote the 4-class rank of K in the narrow sense; i.e., RK= rank ~Kc z = dimv2 ttc'Z/t~41~K/~KJ" Here F 2 is the finite field with two elements, and 2 4 CK/C K is an elementary abelian 2-group which we are viewing as a vector space o v e r F 2. Given a quadratic field K, one can compute R K by computing the rank (over F2) of a certain matrix of Legendre symbols (cf. [11]). Now assume K is imaginary quadratic. So K=Q(1/~mm), where m is a square-free positive integer. For each positive integer t, each nonnegative integer e, and each positive real number x, we define