Resumen de 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

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous

• 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.