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

被引：0

作者：

Xi Mei WU Qin YUE Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjing PRChina State Key Laboratory of Information Security Graduate School of Chinese Academy of SciencesBeijing PRChina ^{[210016
,100039
]}

机构：

来源：

ActaMathematicaSinica(EnglishSeries) | 2007年 / 23卷 / 11期

关键词：

class group; Rédei matrix; reciprocity law;

D O I：

暂无

中图分类号：

O153.4 [域论];

学科分类号：

070104 ;

摘要：

<正> Let F=Q(（-p1p2）1/2) be an imaginary quadratic field with distinct primes p1≡p2≡1 mod8 and the Legendre symbol (（p1）/（p2）)=1.Then the 8-rank of the class group of F is equal to 2 if and onlyif the following conditions hold:（1） The quartic residue symbols (（p1）/（p2）)4=(（p2）/（p1）)4=1;（2） Either bothp1 and p2 are represented by the form a2+32b2 over Z and p2h+（2p1）/4=x2-2p1y2 x,y ∈ Z,or bothp1 and p2 are not represented by the form a2+32b2 over Z and p2h+（2p1）/4=ε（2x2-p1y2）,x,y ∈ Z,ε∈{4-1},where h+（2p1） is the narrow class number of Q(（2p1）1/2).Moreover,we also generalize theseresults.

引用

页码：2061 / 2068

页数：8

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