摘要
设D1、D2是适合D1≥1,D2>1,gcd(D1,D2)=1的无平方因子正整数,IK是虚二次域K=Q(-D1D2)的理想类群。本文证明了:如果存在正整数a、b、k以及奇素数p,可使D1a2+D2b2=kp,gcd(a,b)=1,b∈Np,2k,其中Np是所有不含2tp±1之形素因数的正整数的集合,则当p>5且max(D1,D2)≥1010197时。
Let D 1,D 2 be positive integers such that D 1≥1,D 2>1, gcd (D 1,D 2)=1 and D 1D 2 is square free, and let I k denote the ideal class group of the imaginary quadratic field K=Q(-D 1D 2). Let p be an odd prime ,and let N p be the set of positive integers n such that n has no prime factor q with q≡±1 (mod p). In this paper we prove that if there exist positive integers a,b,k such that D 1a 2+D 2b 2=k p, gcd (a,b)=1, b∈N p,p>5,2k and max (D 1,D 2)≥10 10 197 , then I k has a cyclic subgroup of order p.
出处
《数学杂志》
CSCD
1997年第1期69-71,共3页
Journal of Mathematics
基金
国家自然科学基金
广东省自然科学基金
关键词
虚二次域
理想类群
循环子群
丢番图方程
Imaginary quadratic field, Ideal class group, Cyclic subgroup