代数整数环的素元及剩余类环

The elements of the algebra integer ring——Z[ω]and the remaining kinds rings

作为抽象代数中环理论的两个重要环Z[i]与Z[ω],常以特例的形式散见于抽象代数教材中,对其系统的讨论不多见.而这两个环不仅是抽象代数中的重要实例,而且它们的性质是数论中相关理论的重要基础,特别是Z[ω]在解决费马问题n=3的情形时发挥了关键的作用.文章较为系统的讨论了整环Z[ω],确定了Z[ω]中的素元及其剩余类环所含元素的个数,由此得到数论中一个与Fermat小定理类似的结果。

As two important rings in abstractive algebra, Z[i] and Z[ω] are usually scattered in textbooks as special examples. There is little systematical research on them. t towever, they are not only im- portant real instances in abstractive algebra, at the same time, they are important basis for some concerning theories in count view. The Z[ω]ring plays a key in solving Fei Ma. N = 3. In this article, the author tries to systematically discuss the elements of the integer ring Z[ω], and how many elements are there for its remaining kinds of rings. In this way, the author got a similar theorem with Fermat in count view.