On the 2-class field tower of \(\mathbb Q (\sqrt{2p_1p_2},i)\) and the Galois group of its second Hilbert 2-class field
Given two prime numbers \(p_1\) and \(p_2\) such that \(p_1\equiv p_2 \equiv 5 \mod 8\), the authors study the 2-Hilbert class field tower of the biquadratic extension \(K=\mathbb{Q}(\sqrt{2p_1p_2}, i)\). For a field \(K\), denote by \(K'\) the 2-Hilbert class field of \(K\). The main result of the paper is that the 2-class field tower of \(K\) has length 2, and the Galois group \(G=\mathrm{Gal}((K')'/K)\) satisfies \(|G|= 2h(L) \geq 64,\) where \(L=\mathbb{Q}(\sqrt{2}, \sqrt{p_1p_2}, i).\) The authors also describe the cases where the equality holds. A complementary result is a description of the 2-part of the class group of unramified quadratic extensions of \(K\). The tools used in the proof include the class number formula, as well as reciprocity laws for the Hilbert symbol. Previous results of the authors on class numbers of biquadratic fields are also extensively used.
scientific article; zbMATH DE number 6386912