This report explores several concepts in abstract algebra, including units, irreducibles, norms, division and Euclidean domains before finishing with unique factorisation. All theorems and results that arise from this exploration are proven in full, from a fairly fundamental level. The focus of the report will be on proving Fermat's theorem: that an odd integer prime can be written as the sum of two squared integers if and only if it is congruent to one modulo four. The main sets of interest in this report are the integers and the Gaussian integers. However, in the interests of abstraction, efforts are made to isolate key properties and results, and apply them to a wider context.