Divisibility by 16 of class number of quadratic fields whose 2-class groups are cyclic. (English) Zbl 0535.12002

Yamamoto, Yoshihiko

Osaka J. Math. 21, 1-22 (1984).

Let \(K={\mathbb{Q}}(\sqrt{D})\) be the quadratic field with discriminant D. The strict class number of K is denoted by \(h^+(D)\). In this paper the author considers those fields K for which \(| D|\) has just two distinct prime divisors so that the Sylow 2-subgroup of the strict ideal class group of K is cyclic. Using class field theory he derives conditions for \(h^+(D)\) to be divisible by 16. One of the author’s results is the following: if q is a prime \(\equiv 1 (mod 8)\) such that 8\(| h(-4q)\) then 16\(| h(-4q)\) if and only if \(T\equiv q-1 (mod 16)\), where \(T+U \sqrt{q}>1\) is the fundamental unit of \({\mathbb{Q}}(\sqrt{q})\). This theorem is due to the reviewer [Acta Arith. 39, 381-398 (1981; Zbl 0393.12008)]. Previously it had only been proved by analytic means.

Reviewer: K.S.Williams

MSC:

11R11	Quadratic extensions
11R23	Iwasawa theory

Keywords:

quadratic field; strict class number