| 也谈实二次域类数的可除性 |
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| 引用本文： | 袁平之.也谈实二次域类数的可除性[J].数学学报,1998,41(3):525-530. |
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| 作者姓名： | 袁平之 |
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| 作者单位： | 四川大学数学系,长沙铁道学院数力系 |
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| 摘 要： | 设ｄ无平方因子，ｈ（ｄ）是二次域Ｑ（ｄ）的类数，本文证明了：若１＋４ｋ２ｎ＝ｄａ２，ａ，ｋ＞１，ｎ＞２为正整数，且ａ＜０．９ｋ３５ｎ或ｎ的奇素因子ｐ和ｋ的素因子ｑ均适合（ｐ，ｑ－１）＝１，则除（ａ，ｄ，ｋ，ｎ）＝（５，４１，２，４）以外，ｈ（ｄ）≡０（ｍｏｄｎ）．同时，我们猜测：上述结果中的条件（ｐ，ｑ－１）＝１是不必要的．
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| 关 键 词： | 类数，实二次域，分圆域，完全分裂，丢番图方程 |
| 收稿时间： | 1996-4-12 |
| 修稿时间： | 1997-6-10 |
A Talk about the Divisibility of the Class Number of Real Quadratic Fields |
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| Institution: | Yuan Pingzhi (Department of Mathematics ,Sichuan University, Chengdu 610064, China) (Department of Mathematics and Mechanics, Changsha Railway Institute, Changsha 410075, China) |
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| Abstract: | Let d be a positive square free number, h(d) the class number of the field Q(d). In this paper, we prove that: if a,n>2, k>1 are positive integers with 1+4k 2n =da 2 and a<0.9k 25n or that every prime factor p of n and q of k satisfies (p,q-1)=1. Then h(d)≡0 ( mod n) with the only exception (a,d,k,n)=(5,41,2,4). Meanwhile, we conjecture that the condition (p,q-1)=1 of the above result is not needed. |
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| Keywords: | Class number Real quadratic fields Cyclotomic fields Complete factorizations Diophantine equations |
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