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 Lokokatz3nkl0 September 10th, 2012 04:25 AM

Unique factorization domain

Hello!

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Exercise:
Show that the equation $x^2 + 61= y^3$ has integral solutions. Deduce that $\mathbb{Z}[\sqrt{-61}]$ is not a
unique factorization domain.
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It over a year ago that I had my Algebra lecture, and so I forgot a lot of it....
Also I never had ring-theory very detailed.
So I hope someone can tell me some hints how to show that something is (or in this case isn't) an unique factorization domain!
I looked up the definition, but as I never really worked with it, I don't know how to start...

The Integer solution to the equation are $x= 8$ and $y= 5$.

Thanks and all the best,

Loko

 Lokokatz3nkl0 September 10th, 2012 05:12 AM

Re: Unique factorization domain

I figured a little more out ;)

So I have with this equation that $5^3= (8-\sqrt{-61})(8+\sqrt{-61})$ and thus if $5, (8-\sqrt{-61})$ and $(8+\sqrt{-61})$ are irreducible, the representation of 125 is not unique... right?

If it is right, I would just need help how I could shor the elements to be irreducible.

Thanks and best,

Loko

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