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Math Help - Euclidean Domain

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

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

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