Unique factorization in quadratic fields

Abstract

We need the following definitions: An integral domain is a commutative unitary ring with no zero divisors. A principal ideal domain (PID) is an integral domain in which every ideal can be generated by one element. A unique factorization domain (UFD) is an integral domain in which factorization of integers into primes is unique. (more details later). An integral domain R is said to be a Euclidean ring if for every a≠0 in R there is a defined integer d(a) such that: For all a,b € R, both non zero, d(a) ≤d(ab). For all a,b € R, both non zero, there exists t,r € R such that a=tb+r where either r=0 or d(r) < d(b). A subfield of Є is called a Euclidean field if its set of integers (to be defined later) is a Euclidean ring. Let n be an integer.n is said to be square free if n is not divisible by the square of an integer. Note that: Every Euclidean ring is a PID, but the converse is false. Every PID is a UFD, but the converse is false.