Necessary condition and sufficient for certain Galois group to be metacyclic
The authors consider the biquadratic fields \(K = \mathbb Q(i,\sqrt{d}\,)\), their second Hilbert \(2\)-class fields \(K^2\), and the corresponding Galois groups \(G = \text{Gal}(K^2/K)\). They show that if \(d = 2p\) for some prime \(p \equiv 1 \bmod 4\), then the genus class field \(K^* = \mathbb Q(i,\sqrt{2},\sqrt{p}\,)\) of \(K\) has a \(2\)-class group with rank \(2\) or \(3\), and rank \(3\) if and only if \((2/p)_4 = (p/2)_4 = +1\); here \((p/q)_4\) denotes the quartic residue symbol for primes \(p, q\) satisfying \((p/q) = +1\). Next they show that the Hilbert \(2\)-class field tower of \(K\) terminates after the first step if and only if \((2/p)_4 \neq (p/2)_4\). Finally the prove that \(G\) is a nonabelian metacyclic group if and only if \((2/p)_4 = (p/2)_4 = -1\).
scientific article; zbMATH DE number 5554661