Application of the Euler Phi Function in the Set of Gaussian Integers

Abstract

Integral domains behave mathematically like the set of integers, and as a result, mathematicians are often interested in applying properties and functions of Z to other integral domains in order more thoroughly analyze these sets. One such integral domain is the set of Gaussian Integers, denoted Z[i]. The set of Gaussian Integers shares many characteristics with the set of integers, and it has multiple algebraic and number theoretic applications, including identification of pythagorean triples. In addition to being an integral domain, Z[i], like Z, is further classified as both a unique factorization domain and a euclidean domain. Along with the Gaussian Norm Function, these classifications allow us to analyze relative primality within the set. Because of this, the Euler-o function, which has historically been applied to Z, can also be analyzed within the set of Gaussian Integers.