关于整除等概念在有理数域上的推广

Extension of Concepts of Exact Dividing in the Field of Rational Number

本文将整数环Z中整除、最大公约数、最小公倍数等重要概念合理地引入到有理数域Q中。证明了两个有理数的最大公约数和最小公倍数的存在性,同时给出了一种简单、初等的求法。文中还得出了关于有理数整除、最大公约数、最小公倍数的一些基本性质;并给出了关于“素元”、“互素”等概念和唯一因子分解定理仍然只能在整数环Z中建立与讨论的结论。最后列举了本文所引入的概念与得到的结论的一些简单应用。

This paper introduces rationally the concepts of exact dividing in the ring of integer, greatest common divisor, least common multiple into the field of rational number Q. It proves exitence of two rational numbers--greatest common divisor and least common multiple. It in the meantime, gives a simple and fundamental solution. This paper also comes to conclude some fund amental qualities of exact dividing of rational number, greatest common divison and least common multiple, giving the concepts of prime element, of relative prime and unigue factorization theorem of being only set up and discussed in the ring of integer. In the end, it lists simple applications of the concepts and theorems set in the paper.