Fundamental unit system and class number of real bicyclic biquadratic number fields. (English) Zbl 1011.11071

Wang, Kunpeng

Proc. Japan Acad., Ser. A 77, No. 9, 147-150 (2001).

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 \(D_1= n^2+d\), \(D_2= m^2n^2+r\).

Reviewer: Hideo Yokoi (Aichi)

MSC:

11R20	Other abelian and metabelian extensions
11R27	Units and factorization
11R29	Class numbers, class groups, discriminants

Keywords:

real bicyclic biquadratic number fields; class numbers; fundamental units

References:

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