I need to determine whether or not [itex]\mathcal{O}_{-10} = \mathbb{Z}[\sqrt{-10}][/itex] is a unique factorization domain.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

I just finished proving that [itex]\mathcal{O}_{-5} = \mathbb{Z}[\sqrt{-5}][/itex] 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

[tex]21 = 3\cdot 7 = (1+2\sqrt{-5})(1-2\sqrt{-5})[/itex]

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.

Ill let you guys muse over this for a while.

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# Homework Help: Factorization Problem

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