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 February 3rd, 2012, 12:41 PM #1 Newbie   Joined: Feb 2012 Posts: 1 Thanks: 0 Why is Z[??n] not a UFD for n>2? Hi guys, Could anyone help me with the following problem For n>2 show that $\mathbb{Z}[sqrt{-n}]= \{a + b\sqrt{-n} \|\ a, b \in \mathbb{Z}\}$ is NOT a Unique Factorisation Domain(UFD) I have managed to show 2 is irreducible in this ring for all n>2. Now If we firstly consider the case for n odd, this means n+1 is even and so $2 | (1+n)$. But $(1+n)= (1- \sqrt{-n})(1 + \sqrt{-n})$ and 2 does not divide $(1 - \sqrt{-n})$ or $(1 + \sqrt{-n})$ so 2 is NOT prime. However in a UFD all primes and irreducibles are the same so $Z[\sqrt{-n}]$ cannot be a UFD for when n is odd and bigger than 2. For n even though I don't know how to prove it? I have tried to construct a similar argument as that for n odd but it does not seem to work? I have also tried to factor n in two different ways but again am getting nowhere with it. Any help would be very much appreciated!
 October 2nd, 2014, 08:43 PM #2 Newbie   Joined: Oct 2014 From: USA Posts: 2 Thanks: 2 ℤ[√±n] is not a UFD for non-square n > 2 @Alex25, Let p ∈ ℕ be a prime factor of n(n + 1) = n² + n. E.g. p can be taken as 2. p is irreducible in ℤ[√-n], using Norm properties n² + n = (n + √-n)(n - √-n) -------- (1) p∣ left side of (1) but ∤ either factor on the right, so it is not a prime. ℤ[√-n] cannot be UFD, since it has an irreducible which is a non-prime. Continuing the premise that non-square n >2, considering ℤ[√n], and using similar arguments for n² - n = (n + √n)(n - √n) ℤ[√n] cannot be UFD for non-square n > 2. Thanks from Deveno and topsquark
 October 3rd, 2014, 12:45 PM #3 Newbie   Joined: Oct 2014 From: USA Posts: 2 Thanks: 2 My previous post may be inaccurate Alex's question is theorem 4 in http://www.ias.ac.in/resonance/Volum.../0072-0079.pdf It's an exercise in Michael Artin's book, Algebra When the radicand is positive, the answer is more complicated Per quali d l’anello Z[√d] è UFD, PID, ED? | Carlo Mazza I do not know how to edit a previous article, being new to writing math articles, using unicode, etc. ...Will continue to research on this intriguing question ...

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