Sur un problème de capitulation du corps ℚ $\sqrt {p_1 p_2 } $ , i dont le 2-groupe de classes est élémentaire

Abstract

Let p1 ≡ p2 ≡ 1 (mod 8) be primes such that \(\left( {\tfrac{{p_1 }} {{p_2 }}} \right) = - 1\) and \(\left( {\tfrac{2} {{a + b}}} \right) = - 1\), where p1p2 = a2+b2. Let \(i = \sqrt { - 1} \), d = p1p2, \(\Bbbk = \mathbb{Q}(\sqrt {d,} i),\Bbbk _2^{(1)} \) be the Hilbert 2-class field and \(\Bbbk ^{(*)} = \mathbb{Q}(\sqrt {p_1 } ,\sqrt {p_2 } ,i)\) be the genus field of \(\Bbbk \). The 2-part \(C_{\Bbbk ,2} \) of the class group of \(\Bbbk \) is of type (2, 2, 2), so \(\Bbbk _2^{(1)} \) contains seven unramified quadratic extensions \(\mathbb{K}_j /\Bbbk \) and seven unramified biquadratic extensions \(\mathbb{L}_j /\Bbbk \). Our goal is to determine the fourteen extensions, the group \(C_{\Bbbk ,2} \) and to study the capitulation problem of the 2-classes of \(\Bbbk \).