Arithmetic in the ring of Gaussian integers

Abstract

We study the ring of integers Z, and use it’s properties along with those of complex numbers to explore the nature of the ring of Gaussian integers Z[i]. We introduce an analogue of the key concepts of the ring Z in Z[i] such as factorization and modular arithmetic. One approach is by mimicking the statements and proofs in Z and modifying them to accommodate the 2-dimensional aspect of the elements of Z[i]. Then we investigate ways of extending this information to general quadratic rings of the form Z [√d], whered is a square-free integer.