Units and 2-class field towers of some multiquadratic number fields. (English) Zbl 1455.11140

Chems-Eddin, Mohamed Mahmoud; Zekhnini, Abdelkader; Azizi, Abdelmalek

Turk. J. Math. 44, No. 4, 1466-1483 (2020).

Extending some former studies on quadratic and biquadratic number fields (see for instance [E. Benjamin and C. Snyder, Math. Scand. 76, No. 2, 161–178 (1995; Zbl 0847.11058); A. Azizi and I. Benhamza, Ann. Sci. Math. Qué. 29, No. 1, 1–20, (2005; Zbl Zbl.1217.11097)]), the authors of the present paper investigate the \(2\)-class field towers of some multiquadratic number fields related to the imaginary triquadratic field \(\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{d})\) whose \(2\)-class group is of type \((2,2),\) where \(d\) is an odd squarefree natural number. Specifically, they characterize the \(2\)-class groups of certain imaginary multiquadratic fields, and they determine unit groups of some multiquadratic fields of degree \(8\) and \(16.\) This allows them to study the \(2\)-class field tower of certain families of imaginary triquadratic fields and deduce the capitulation behaviors in the unramified quadratic extensions of such fields.

Reviewer: Toufik Zaïmi (Riyadh)

MSC:

11R04	Algebraic numbers; rings of algebraic integers
11R11	Quadratic extensions
11R27	Units and factorization
11R29	Class numbers, class groups, discriminants
11R39	Langlands-Weil conjectures, nonabelian class field theory

Keywords:

multiquadratic number fields; unit groups; 2-class groups; Hilbert 2-class field towers

Citations:

Zbl 0847.11058