Galois module structure of units in real biquadratic number fields. (English) Zbl 1060.11070

Mazur, Marcin; Ullom, Stephen V.

Acta Arith. 111, No. 2, 105-124 (2004).

Let \(p\) and \(q\) be primes and consider the bicyclic number field \(N = \mathbb Q(\sqrt{p},\sqrt{q}\,)\). If the quadratic subfield \(K = \mathbb Q(\sqrt{pq}\,)\) has elementary abelian \(2\)-class group, then the structure of the unit group of \(N\) can easily be described in terms of the fundamental units of the quadratic subfields of \(N\). In the case where the class number \(h(K)\) is divisible by \(4\), the authors construct the corresponding governing field for divisibility by \(8\) [cf. P. Stevenhagen’s thesis, Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1988/89, No. 1, 93 p. (1989; Zbl 0701.11056)] and give the structure of the unit group of \(N\) for those fields for which \(h(K)\) is not divisible by \(8\).
In a second part, the authors study the existence of Minkowski units in general bicyclic biquadratic number fields.

Reviewer: Franz Lemmermeyer (Bilkent)

MSC:

11R16	Cubic and quartic extensions
11R29	Class numbers, class groups, discriminants

Keywords:

unit groups; biquadratic number fields; class group; governing field

Citations:

Zbl 0701.11056