8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Abstract

Let \( F = \mathbb{Q}{\left( {{\sqrt { - p_{1} p_{2} } }} \right)} \) be an imaginary quadratic field with distinct primes p1 ≡ p2 ≡ 1 mod 8 and the Legendre symbol \( {\left( {\frac{{p_{1} }} {{p_{2} }}} \right)} = 1 \). Then the 8-rank of the class group of F is equal to 2 if and only if the following conditions hold: (1) The quartic residue symbols \( {\left( {\frac{{p_{1} }} {{p_{2} }}} \right)}_{4} = {\left( {\frac{{p_{2} }} {{p_{1} }}} \right)}_{4} = 1 \); (2) Either both p1 and p2 are represented by the form a2 +32b2 over ℤ and \( p^{{h_{ + } {\left( {2p_{1} } \right)}/4}}_{2} = x^{2} - 2p_{1} y^{2} ,x,y \in \mathbb{Z} \), or both p1 and p2 are not represented by the form a2 +32b2 over ℤ and \( p^{{h_{ + } {\left( {2p_{1} } \right)}/4}}_{2} = \varepsilon {\left( {2x^{2} - p_{1} y^{2} } \right)},\;x,y \in \mathbb{Z},\;\varepsilon \in {\left\{ { \pm 1} \right\}} \), where h+(2p1) is the narrow class number of \( \mathbb{Q}{\left( {{\sqrt {2p_{1} } }} \right)} \). Moreover, we also generalize these results.