Unique prime factorization in imaginary quadratic number fields
Unique prime factorization in imaginary quadratic number fields
复制标题
虚二次数域中唯一的素因数分解
DOI：
10.1007/bf01904852
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发表时间：
1965
期刊：
影响因子：
--
通讯作者：
E. Lanczi
中科院分区：
文献类型：
--
作者：
E. Lanczi
众所周知，如果P表示有理域，则可以在P（l/d-）的理想类和所提到的二次型的类之间进行一一对应。因此，海尔布龙和LINFOO~的定理意味着，在上述九种情况下，P（l/d）的类数是l，并且这最多对于另一个负d是可能的。回想一下，如果理想类的数量是1，那么算术基本定理对P（l/d）的整数成立。因此，在上述九个d的情况下，P（l/d）的整数形成唯一的因子分解域。如果ld]-<-11，则所述域是欧几里得域。另一方面，已知在虚二次域的整数中不存在更多的欧氏环。因此，在前五种情况下，欧几里德算法是证明算术基本定理的自然工具。但在后四种情况下，据我所知，不存在任何不利用理想类的证明。1
It is known, that if P denotes the rational field, then one can make a one-to-one correspondence between the ideal classes of P (l/d-) and the classes of the mentioned quadratic forms. Thus the theorem of HEILBRONN and LINFOO~ implies that in the mentioned nine cases the class number of P (l/d) is l, and this can be possible at most for one further negative d.Recall that if the number of the ideal classes is 1, then the fundamental theorem of arithmetic holds for the integers of P (]/)-). Thus in the case of the mentioned nine d, the integers of P (l/d) form a unique factorization domain. If ld]-<-11, the mentioned domains are Euclidean. On the other hand, it is known that there does not exist more Euclidean rings among the integers of imaginary quadratic fields. Thus in the first five cases the Euclidean algorithm is a natural tool for proving the fundamental theorem of arithmetic. But in the last four cases,--as far as I know--there does not exist any proof which does not make use of the ideal classes. 1