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 p ₁ ≡ p ₂ ≡ 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 p ₁ and p ₂ are represented by the form a ² +32b ² over ℤ and $p^{{h_{ + } {\left( {2p_{1} } \right)}/4}}_{2} = x^{2} - 2p_{1} y^{2} ,x,y \in \mathbb{Z}$, or both p ₁ and p ₂ are not represented by the form a ² +32b ² 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 ₊(2p ₁) is the narrow class number of $\mathbb{Q}{\left( {{\sqrt {2p_{1} } }} \right)}$. Moreover, we also generalize these results.