A quick route to unique factorization in quadratic orders

TL;DR

This paper presents a concise proof demonstrating when quadratic orders have unique factorization, avoiding traditional methods like ideal class groups or geometry of numbers.

Contribution

It provides a novel, simplified proof of the criterion for unique factorization in quadratic orders, bypassing classical approaches.

Findings

Short proof of the criterion for unique factorization

Avoids reliance on ideal class groups

Simplifies understanding of quadratic order factorizations

Abstract

We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.