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Thread: Euclidean Domain

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Euclidean Domain

    Suppose $\displaystyle \rho \in \mathbb{Z}[i] $ and that $\displaystyle \lambda(\rho) = r $ is prime in $\displaystyle \mathbb{Z} $. Show that $\displaystyle \rho $ is prime in $\displaystyle \mathbb{Z}[i] $.

    I think the easiest way is to use the contrapositive, but I'm not sure.
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    Suppose $\displaystyle \rho \in \mathbb{Z}[i] $ and that $\displaystyle \lambda(\rho) = r $ is prime in $\displaystyle \mathbb{Z} $. Show that $\displaystyle \rho $ is prime in $\displaystyle \mathbb{Z}[i] $.

    I think the easiest way is to use the contrapositive, but I'm not sure.
    If $\displaystyle \rho = \rho_1 \rho_2$ for non-units $\displaystyle \rho_1,\rho_2$ then $\displaystyle r=\lambda (\rho) = \lambda (\rho_1)\lambda (\rho_2)$. This is a contradiction. Can you see why?
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