Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields (Q6591982)

For a number field \(k\), denote by \(k_2^{(1)}\), the Hilbert \(2\)-class field of \(k\); that is, the maximal unramified, abelian extension of degree a power of \(2\), over \(k\). Write \(k_2^{(0)}\) for \(k\), and \(k_2^{(i+1)}\) for the Hilbert \(2\)-class field of \(k_2^{(i)}\) for each \(i \geq 0\). It is an unresolved problem to determine when this \(2\)-class field tower is infinite. When the rank of the \(2\)-Sylow subgroup \(\mathrm{Cl}_2(k_2^{(1)})\) is \(\leq 2\), \N\textit{N. Blackburn} showed in [Proc. Camb. Philos. Soc. 53, 19--27 (1957; Zbl 0077.03202)] that the tower has finite length \(\leq 3\). Further, in 1980, \textit{B. Schmithals} [Arch. Math. 34, 307--312 (1980; Zbl 0448.12008)] constructed imaginary quadratic fields \(k\) with infinite \(2\)-class field tower, if the rank of \(\mathrm{Cl}_2(k_2^{(1)})\) is \(3\). Hence, it is an important problem to determine all the fields for which the last-mentioned rank is \(\leq 2\).\N\NA project started by E. Benjamin and others more than two decades ago, aims to characterize all quadratic fields with the above rank condition. The present paper also falls in line with this project.\N\NHere, the authors consider real quadratic fields \(k\), and aim to study when \(\mathrm{Cl}_2(k_2^{(1)})\) is cyclic, under the assumption that \(\mathrm{Cl}_2(k)\) has the type \((2^n, 2^m)\) for some \(n \geq 1, m \geq 2\), and the discriminant of \(k\) is divisible by some primes congruent to \(3\) modulo \(4\). More precisely, they consider the fields \(k = \mathbb{Q}(\sqrt{2p_1p_2q})\) where \(p_1,p_2\) are primes \(\equiv 1\) mod \(4\) and, \(q\) is a prime \(\equiv 3\) mod \(4\), where \(\mathrm{Cl}_2(k)\) is supposed to have type \((2,2^n)\) with \(n \geq 2\). They obtain precise criteria for \(\mathrm{Gal}(k_2^{(2)}/k)\) to be metacyclic, and if it is not metacyclic, they obtain precise criteria for \(\mathrm{Cl}_2(k_2^{(1)})\) to be a cyclic group. Their work is a continuation of their earlier work [Int. J. Number Theory 15, No. 4, 807--824 (2019; Zbl 1457.11151)].