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 ?

Every PID is UFD, and in a PID every irreducible is prime. – Hassan Muhammad Nov 18 '11 at 16:25 — Hassan Muhammad, Nov 18 '11 at 16:25

score 13 · Accepted Answer · answered Nov 18 '11 at 16:17

13

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.

answered Nov 18 '11 at 16:17

Dylan Moreland

19,819

Thank you ! It's rather what I want . – Mahan Nov 18 '11 at 17:31

score 6 · Answer 2 · answered Nov 18 '11 at 17:15

6

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.

answered Nov 18 '11 at 17:15

interesting theorem ! thanks for your useful comment . – Mahan Nov 18 '11 at 17:29
You're most welcome. – Nov 18 '11 at 17:31

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

2 Answers2