On Algebraic Numbers and Quadratic Extensions

Abstract: Abstract. In this thesis some basic properties of algebraic el- ements, integral elements and extensions of rings and elds are investigated. Special focus is given to quadratic extensions. It is assumed that the reader is familiar with some elementary concepts from abstract algebra, such as the denitions of rings and elds. The rst section covers the basics of algebraic elements and in- tegral elements and serves as a foundation for the rest of the thesis. In section two and three some theory of Unique Factorization Do- mains and Euclidean Domains is developed. The concepts and results from these two sections are used in section four and ve, which treat Quadratic Extensions and certain Ring Extensions of the integers.