- #1
Oxymoron
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I need to determine whether or not [itex]\mathcal{O}_{-10} = \mathbb{Z}[\sqrt{-10}][/itex] 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 [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.
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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