On the 2-class field tower of $$\mathbb Q (\sqrt{2p_1p_2},i)$$ Q ( 2 p 1 p 2 , i ) and the Galois group of its second Hilbert 2-class field

Abstract

Let \(p_1\) and \(p_2\) be primes such that \(p_1\equiv p_2\equiv 5 \pmod 8\), \(i=\sqrt{-1}\), \(d=2p_1p_2\), \(\mathbb K =\mathbb Q (\sqrt{d},i)\), \(\mathbb K _2^{(1)}\) be the Hilbert 2-class field of \(\mathbb K \), \(\mathbb K _2^{(2)}\) be the Hilbert 2-class field of \(\mathbb K _2^{(1)}\), \(G\) be the Galois group of \(\mathbb K _2^{(2)}/\mathbb K \) and \(\mathbb K ^{(*)}=\mathbb Q (\sqrt{p_1},\sqrt{p_2},\sqrt{2}, i)\) be the genus field of \(\mathbb K \). The 2-part \(\mathbf C _{\mathbb{K },2}\) of the class group of \(\mathbb K \) is of type \((2, 2, 2)\). Our goal is to study the 2-class field tower of \(\mathbb K \) and to calculate the order of \(G\).