Divisibility by 16 of class number of quadratic fields whose 2-class groups are cyclic
Yoshihiko Yamamoto · Osaka City University (Osaka City University) · 1984
Introduction.Let K=Q(\/J)) be the quadratic field with discriminant Z), and H(D) and h(D) be the ideal class group of K and its class number respectively.The ideal class group of K in the narrow sense and its class number are denoted by H + (D) and h + (D) respectively.We have h + (D)=2h(D) > if D>0 and the fundamental unit £ D (>1) has the norm 1, and h + (D) = h(D), otherwise.We assume, throughout the paper, that | D | has just two distinct prime divisors, written p and q, so that the 2-class group of K (i.e. the Sylow 2-subgroup of H + (D) because we mean in the narrow sense) is cyclic.Then the discriminant D can be written uniquely as a product of two prime discriminants d λ and d 2y D=d ι d 2y such that p\d λ and q\d 2 (cf.[16], for example).By Redei and Reichardt [13] (cf.proposition 1.2 below), h + (D) is divisible by 4 if and only if D belongs to one of the following 6 types: