An Algorithm for Determining (A,B) in Certain Euclidean Domains

The purpose of this paper is to study the concept of a greatest common divisor in such a way that it may be used as an enrichment topic at the senior high school level. In order to do this, the concept has been approached from both a theoretic and an applied point of view. In the theoretical areas, the algebraic structures involved are explored, along with the Euclidean algorithm, which provides a means for computing the greatest common divisor of two elements in a Euclidean ring and expressing it as a linear combination of these elements. In the more applied sections, such an algorithm is programmed for a computer. It is used in a BASIC program which will compute (a, b) in the ring of polynomial forms over a field and express it as a linear combination of a and b. Some other Euclidean rings, which the teacher might find instructive to explore with a class, are also discussed.