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 September 10th, 2012, 04:25 AM #1 Newbie   Joined: Sep 2012 Posts: 2 Thanks: 0 Unique factorization domain Hello! --------------------- 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. --------------------- 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
 September 10th, 2012, 05:12 AM #2 Newbie   Joined: Sep 2012 Posts: 2 Thanks: 0 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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