On Quadratic Extensionsand Gaussian Primes

Abstract: Abstract. This thesis will deal with algebraic extensions. The goal is to give the reader an introduction to algebraic extensions, euclidian domains, unique factorization domains as well as more specic theories for example how to nd primes in the gaussian integers. The layout of the report will be to start with the more general theories and as we progress narrow down on more specic theories. It is assumed that the reader already is familiar with some basic algebra. The rst two chapters will introduce the reader to algebraic ex- tensions and some more general denitions of primes and other properties. The next three chapters will cover some more speci- k properties of rings such as unique factorization. In the last three chapters we will move on to investigate more specik exten- sions starting with quadratic extensions and then moving on to the gaussian integers as well as introducing the reader to some un- solved problems in the nal chapter. Throughout this whole thesis there will be a lot of examples where an example following a denition or theorem will try to illustrate that particular concept.