On the units and the class numbers of certain composita of two quadratic fields (Q1302090)

The authors intend to obtain some results on the class number \(h_K\) and the unit group \(E_K\) of the real biquadratic field \(K\), which is the composita of a real quadratic field \(k_1\) and the real quadratic field \(k_2\) derived from \(k_1\). Namely, they assert that for each fundamental unit \(\varepsilon_1, \varepsilon_2, \varepsilon_3\) and each class number \(h_1, h_2, h_3\) of three real quadratic subfields \(k_1, k_2, k_3\) of \(K\), \(\{\varepsilon_1, \varepsilon_2, \varepsilon_3\}\) is a fundamental system of units of \(K\) and \(h_K= \frac 14\cdot h_1\cdot h_2\cdot h_3\) holds except for computable finite number of quadratic fields. However, this fact is true only if \(k_1\) is such a quadratic field of special Richaud-Degert type that \(k_1= \mathbb{Q} (\sqrt{M^2\pm 4})\) for an integer \(M\) [cf. for example, the reviewer's paper, Nagoya Math. J. 33, 139-152 (1968; Zbl 0167.04401)].