On the second Hilbert 2-class field of real quadratic number fields with 2-class group isomorphic to \((2,2^n)\), \(n\geq 2\) (Q1974415)

The real quadratic number fields \(k\) with abelian \(2\)-class field tower were classified in [the author and \textit{F. Lemmermeyer}, J. Number Theory 73, 182-194 (1998; Zbl 0919.11073)]; here the author assumes that \(k\) has a \(2\)-class group isomorphic to \(\mathbb Z/2 \times \mathbb Z/2^n\), and that the \(2\)-class field tower is nonabelian. He then determines whether the Galois group \(\Gamma\) of the second Hilbert \(2\)-class field of \(k\) is a modular, metacyclic non-modular, or nonmetacyclic \(2\)-group, and the criteria he gives involve capitulation in certain quadratic unramified extensions of \(k\) as well as conditions on fundamental units.