Unique Factorization for Ideals in the Ring 𝑍[√−5]

The fundamental theorem of arithmetic says that any integer greater than 2 can be written uniquely as a product of primes. For the ring 𝒁[√−𝟓], although unique factorization holds for ideals, unique factorization fails for elements. We investigate both elements and ideals of𝒁[√−𝟓]. For elements, we examine irreducibility (the analog of primality) in 𝒁[√−𝟓] and look at how often and how badly unique factorization fails. For ideals, we examine irreducibility again and a proof for unique factorization.