Non-Unique Factorization in \mathbb{Z}[\sqrt{-10}]

[Oxymoron]

I need to determine whether or not \mathcal{O}_{-10} = \mathbb{Z}[\sqrt{-10}] is a unique factorization domain.

Now, I think the short answer is simply: NO.

The question is meant to be simple (I think).

I just finished proving that \mathcal{O}_{-5} = \mathbb{Z}[\sqrt{-5}] is NOT a unique factorization domain and it took me two pages. It involved me finding an integer which had two DIFFERENT factorizations into irreducibles. Using maple and the "factorEQ" command with (numtheory) I found that

21 = 3\cdot 7 = (1+2\sqrt{-5})(1-2\sqrt{-5})[/itex]<br /> <br /> But now for the question at hand, Maple cannot find an integer which equals two factorizations of this sort because, unlike 5, 10 is not a prime.<br /> <br /> Ill let you guys muse over this for a while.

 

[Oxymoron]

I suppose I could just use

26 = 2\cdot 13 = (4+\sqrt{-10})(4-\sqrt{-10})

and show that no element of \mathbb{Z}[\sqrt{-10}] of norm 2 or 13 hence 2 and 13 are irreducible. And then show there is no element of norm 4+\sqrt{-10} or 4-\sqrt{-10} and thus, they are irreducible. If I can show that then I have shown that 26can be written as two DIFFERENT factorizaruib into irreducibles which implies the ring is a UFD. This shouldn't be too hard if its the right thing to do. Is it the right thing to do?