In this thesis we explore the unit group of the ring of integers of number fields. In our exploration we look at Dirichlet’s unit theorem which shows that the unit group is a finitely generated abelian group. This will allow us to explore computing the generating set of the unit group. From this basis we then extend the computation of unit groups to describe and implement an algorithm in PARI for finding elements in the kernel of the norm mapping K×/K×2 Q×/Q×2. Elements in this mapping are of particular interest for finding Brauer Manin obstructions with current implementations using a set of fundamental units.