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

Authors:	Xi Mei Wu Qin Yue

Institution:	(1) Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China;(2) State Key Laboratory of Information Security Graduate School of Chinese Academy of Sciences, Beijing, 100039, P. R. China

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. Project supported by the National Natural Science Foundation of China (No. 10371054)

Keywords:	class group Ré dei matrix reciprocity law
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