This thesis deals with quadratic integer rings, in particular the Gaussian integers Z}[i]. Concepts such as quadratic extensions, Euclidean domains and unique factorization domains will be introduced to the reader. The goal of this thesis is to show how a natural generalization of the integers Z, in the form of the Gaussian integers, can be used to prove important results in Z. Furthermore, it aims to explore other types of quadratic integer rings and their differences.

The first two sections will introduce the Gaussian integers, integral domains and norms. In the third section the irreducible elements of the Gaussian integers are categorized. The fourth and fifth sections talk about general quadratic integer rings and their norms. Section six and seven categorizes the prime elements and introduces unique factorization domains.