# Unique Factorization in the Rings of Integers of Quadratic Fields: A Method of Proof

Spring 4-2023

## School

College of Arts and Sciences

Mathematics

## Keywords

Unique Factorization, Factorization, Quadratic Fields, Proof, Gauss, Class Number Problem

## Disciplines

Algebra | Number Theory

## Abstract

It is a well-known property of the integers, that given any nonzero aZ, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + bD | a, bZ, D a negative, squarefree integer}, we find that the FTA does not always hold. For example, in the domain O(√−5), 6 = 2 · 3 and 6 = (1 + √−5)(1 − √−5) are two valid factorizations of 6, with 2, 3, (1 + √−5), (1 − √−5) all irreducible elements in O(√−5). This paper discusses the history and development of the problem of discerning which fields of the form O(√D) are unique factorization domains (UFDs) and concludes by constructing a method of proving unique factorization in some domain using results concerning Euclidean domains and principal ideal domains.

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