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

Xi Mei Wu; Qin Yue
November 2007
Acta Mathematica Sinica;Nov2007, Vol. 23 Issue 11, p2061
Academic Journal
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 +32 b 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 +32 b 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 +(2 p 1) is the narrow class number of $$ \mathbb{Q}{\left( {{\sqrt {2p_{1} } }} \right)} $$ . Moreover, we also generalize these results.


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