On Integers, Primes and UniqueFactorization in Quadratic Fields

Abstract: Abstract. This thesis will deal with quadratic elds. The prob- lem is to study such elds and their properties including, but not limited to, determining integers, nding primes and deciding which quadratic elds have unique factorization. The goal is to get famil- iar with these concepts and to provide a starting point for students with an interest in algebra to explore eld extensions and inte- gral closures in relation to elementary number theory. The reader will be assumed to have a basic knowledge in algebra and famil- iar with concepts such as groups, rings and elds. The necessary background material is covered in for example A First Course In Abstract Algebra by John B. Fraleigh. Some familiarity with basic number theory may be helpful, but not necessary for the scope of this thesis. The questions posed in this thesis was answered by means of literature and discussions with fellow students and my supervisor. The rst four sections will deal with basic concepts in algebra such as algebraic numbers, algebraic integers and prime numbers. This knowledge will then be applied to the subject of quadratic elds. The thesis is concluded with two sections about important cases of quadratic elds, Gaussian and Eisenstein.