Fundamental unit system and class number of real bicyclic biquadratic number fields
For three certain types of real bicyclic biquadratic number fields, the author intends to determine explicitly the class numbers and the fundamental systems of the unit groups by using the class numbers and the fundamental units of their quadratic subfields. Namely, let \(D_1= n^2+2\), \(D_2= n^4+2n^2+2\), \((2\mid n\geq 4)\), and let \(D_1,D_2\) be square free. Then the class number \(h_K\) and the fundamental system \(U_K\) of the unit groups of \(K= \mathbb{Q}(\sqrt{n^2+2}, \sqrt{n^4+2n^2+2})\) are \(h_K= (\frac 12)h_1h_2h_3\) and \(U_K= \{-1, \varepsilon_1, \varepsilon_2, \sqrt{\varepsilon_1 \varepsilon_3}\}\), where \(h_i\) and \(\varepsilon_i\) mean respectively the class number and the fundamental unit of subfields \(k_i= \mathbb{Q}(\sqrt{D_i})\) for \(i=1,2\), and \(k_3= \mathbb{Q} (\sqrt{D_1D_2})\). The other remainder types are the following: \(D_1= (n^2+rd) (n^2+sd)\), \(D_2= (n^2+td) (n^2+ud)\) and \(D_1= n^2+d\), \(D_2= m^2n^2+r\).
scientific article; zbMATH DE number 1779508