A table of ideal class groups of imaginary quadratic fields
A table of ideal class groups of imaginary quadratic fields
复制标题
虚二次域理想类群表
DOI：
10.3792/pja/1195520300
复制
发表时间：
1970
期刊：
影响因子：
--
通讯作者：
Hideo Wada
中科院分区：
文献类型：
--
作者：
Hideo Wada
下表将显示虚二次域Q(/-m)， 0m24000的所有理想类群，它们在下面解释的意义上不是“平凡的”。理想类群是有限阶的阿贝尔群，用阿贝尔群的基本定理表示为a、b、…阶循环群的直积。， c，我们可以假设aZbZ…cZ。我们用（a, b，…）来表示这样的群。c)。设t为Q（/-m）的有理数分支数。那么，已知a， b，。c是t1，我们有b ch（-类数Q(/-m)）当群的结构平凡地由t和h决定时，即当它在t—1或2时是循环的，或者当它的类型为（a, 2,）时，我们称其为“平凡”群。， 2)如果t >= 3。在该表中，所有组的类型为（a, b）、（a, b, 2）或（a, b, 2, 2）。因此我们可以推测“所有虚二次域的理想类群都是循环的”或“类型为（a,b,，…）”本文利用东京大学数学系的TOSBAC-3000电子计算机进行计算。这个计算需要两个小时的计算机时间。（0 - 24000的h表将在京都大学数学科学研究所的《Sfiri Kaiseki Kenkyfijo KSkyfiroku》报告中发表。）
The following table will show all ideal class groups of imaginary quadratic fields Q(/-m), 0m24000, which are not ’trivial’ in the sense explained below. The ideal class group is an abelian group of finite order, which is expressed, by the fundamental theorem on abelian groups, as the direct product of cyclic groups of orders a, b,..., c, where we can assume that aZbZ... cZ. We shall denote such a group by (a, b, ..., c). Let t be the number of rational primes ramified in Q(/-m). Then it is well-known that the number of even numbers among a, b, ., c is t1 and we have a b. ch (-the class number of Q(/-m)). We shall call the group ’trivial’, when its structure is trivially determined by t and h, i.e. when it is cyclic if t--1 or 2 or when it is of the type (a, 2, ., 2) if t >= 3. In this table, all the groups are of types (a, b), (a, b, 2) or (a, b, 2, 2). So we may conjecture" "All ideal class groups of imaginary quadratic fields will be cyclic 2" 2 2r)." or of type(a,b, ,.. For computing this table, the author used the electoronic computer TOSBAC-3000 installed in the Depertment of Mathematics University of Tokyo. This calculation required two hours computer time. (The table of h for 0m 24000 will be published in a report "Sfiri Kaiseki Kenkyfijo KSkyfiroku" of Research Institute for Mathematical Sciences, Kyoto University.)