二次整数环的迭代图

The Iteration Digraphs of Quadratic Rings

令OK为有理数域Q的二次扩张K=Q(槡d)的代数整数环,pOK是由有理素数p生成的OK的理想.定义商环OK/pOK上的迭代图G(OK,t),t为OK中的元素.迭代图G(OK,t)的顶点为OK/pOK中的所有元素,并且对于图中的两个顶点α和β,如果β=tα,则从α到β有一条有向边.该文根据理想pOK的结构研究迭代图G(OK,t),给出位于同一个圈上的点的相互关系,以及图G(OK,t)的具体形式.

Let OK be the ring of integers of the quadratic field K =Q(d) where Q is the set of rational numbers,and dis asquare-free rational integer.Let pbe arational prime,we denote by pOK the ideal of OK generated by p.For agvien t∈OK,we construct an iteration digraph G(OK,t),whose vertex set is OK/pOKand for which there is adirected edge from α to β∈Rifβ=tα.In this paper,we investigate the structure of the iteration digraphs G(OK,t)in terms of the rational prime p.