Unit, Irreducible, and Prime Elements of The Integral Domain Z[sqrt (5)]

Abstract

In abstract algebra at the undergraduate level, the ring Z[sqrt(5)] is often used as a simple example of an integral domain that does not satisfy the unique factorization domain (UFD) but Z[sqrt(5)] is Halfway Factorial Domain (HFD). Unlike the ring of integers (Z)  or the Gaussian integers (Z[i])  . Z[sqrt(5)] contains elements that admit non-unique factorizations, making it an interesting subject of study. A key challenge in analyzing the structure of  lies in its limited group of units, consisting only of +-1, as well as the existence of irreducible elements that are not necessarily prime. This phenomenon leads to ambiguity in factorization, necessitating a deeper investigation into its arithmetic properties. This research aims to explore the factorization characteristics in Z[sqrt(5)], analyze irreducible elements and their relation to primality, and examine the implications of non-unique factorization on its algebraic structure. The findings are expected to contribute to a more comprehensive understanding of quadratic rings and their applications in number theory.