Is $\mathbb Z[ \sqrt{-3}]$ UFD?

Question

Is $\mathbb Z[ \sqrt{-3}]$ UFD ? If so , why $4=2 \times 2 = (1+\sqrt{-3}) \times (1-\sqrt{-3} )$ and every terms are irreducible ?

Answer 1

I think your suspicion is correct. On the other hand, the integral closure of $\mathbf{Z}$ in $\mathbf{Q}(\sqrt{-3})$ is the larger ring $\mathbf{Z}[\frac{1 + \sqrt{-3}}{2}]$, since $-3 \equiv 1 \bmod{4}$. This is norm-Euclidean and in particular a UFD. Note that all UFDs are integrally closed, so this is another way to see that $\mathbf{Z}[\sqrt{-3}]$ does not have unique factorization.

Answer 2

FYI, $\mathbb{Z}[\sqrt{-3}]$ is not only not a UFD, but it's the unique imaginary order of a quadratic ring of algebraic integers that has the half-factorial property (Theorem 2.3)--ie any two factorizations of a nonzero nonunit have the same number of irreducibles.