대수적 관점에서 본 소수 지도 의의에 관한 연구

Abstract

This study tries to justify learning prime numbers in view of algebra. To get information from a set A or provide A with algebraic structures, we first have to apply an onto function 'f:Z→A' which represents the correspondence from a set of integers Z to set A. Then based on the set theory using the onto function, set A is represented as residues of Z. Finally to change the algebraic structures of Z, arithmetic shifts take place. After this process, set A comes to have a ring structure. To effectively handle the ring, the ring need to be broken up into direct sums or direct products. In this process, we need prime numbers to break them up into the smallest unit. By checking these theoretical backgrounds, the researcher tries to present the necessity of teaching prime numbers, great common divisors, least common multiples, and Euclidean algorithm. Finally, based on the result, this study suggests that, in middle and high schools, the prime numbers should be taught through composing and decomposing on the reversible point of view, that Euclidean algorithm need to be emphasized, that the lessons of great common divisors and least common multiples should be presented based on lattices, and that above all, the concept and existence of prime numbers must be taught first. This study is expected to provide theoretical foundation for the mathematics education.