Class numbers of quadratic extensions of algebraic number fields (Q1086287)

\textit{Y. Yamamoto} [Osaka J. Math. 7, 57-76 (1970; Zbl 0222.12003)] has constructed infinitely many imaginary quadratic fields which have a subgroup of their class group isomorphic to \({\mathbb{Z}}/n\times {\mathbb{Z}}/n\). \textit{H. Naito} [J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 205-211 (1985; Zbl 0582.12004)] has generalized this result to totally imaginary extensions of a given totally real field. In this article the author obtains three results which give infinitely many extensions K of a given field F such that K has a fixed group H as a subgroup of its class group. In the first result K/F is a quadratic extension and H is isomorphic to \({\mathbb{Z}}/n\). For the second result \([K:F]=p\) is an odd prime, F is a quadratic field and H is isomorphic to \({\mathbb{Z}}/n\). In the final result K/F is again quadratic, but H is isomorphic to \({\mathbb{Z}}/3\times {\mathbb{Z}}/3\). In all three results \(N_{K/F}(H)=1\) and some restrictions are placed on the infinite primes of K and F. The proofs employ the diophantine equation which was used in Yamamoto's proof.