Title: How Do Elements Really Factor in $\mathbb{Z}[\sqrt{-5}]$?,
Abstract: Most undergraduate level abstract algebra texts use $\mathbb{Z}[\sqrt{-5}]$
as an example of an integral domain which is not a unique factorization domain
(or UFD) by exhibiting two distinct irreducible factorizations of a nonzero
element. But such a brief example, which requires merely an understanding of
basic norms, only scratches the surface of how elements actually factor in this
ring of algebraic integers. We offer here an interactive framework which shows
that while $\mathbb{Z}[\sqrt{-5}]$ is not a UFD, it does satisfy a slightly
weaker factorization condition, known as half-factoriality. The arguments
involved revolve around the Fundamental Theorem of Ideal Theory.