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