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Math Help - For what NEGATIVE values of d is Z[sqrt[d]] a Euclidean domain?

  1. #1
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    For what NEGATIVE values of d is Z[sqrt[d]] a Euclidean domain?

    I was asked to prove that the ring Z[sqrt[d]] is not euclidean for any d <= -3. (Note - The resulting ring it not contained in R!)

    I proved it for certain d's easily (For example d=-3), by showing that it is not a UFD, but am not sure how to prove it in the more general case. (Or whether Z[sqrt[d]] is a UFD for some d's although it is not Euclidean)

    Could anyone suggest a solution or intuition as to why is it not a Euclidean Domain?
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: For what NEGATIVE values of d is Z[sqrt[d]] a Euclidean domain?

    for Z[\sqrt{-2}] note (1-\sqrt{-2})(1+\sqrt{-2}) = 3 If this were a euclidean domain, it would be a PID, which means primes are irreducible. Since neither of the factors are units, its safe to say this is not a Euclidean domain

    Same reasoning for Z[\sqrt{-1} take (1-\sqrt{-1})(1+\sqrt{-1}) = 2 with neither factors being units.
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