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

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.


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


Zbl 0847.11058
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