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Math Help - integral closure of Z[2i]?

  1. #1
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    integral closure of Z[2i]?

    Find the quotient field K of \mathbb{Z}[2i] and the integral closure of \mathbb{Z}[2i] in K.
    ( i=\sqrt{-1})
    Last edited by mr fantastic; March 21st 2011 at 01:32 AM. Reason: Title.
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  2. #2
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    Quote Originally Posted by KaKa View Post
    Find the quotient field K of \mathbb{Z}[2i] and the integral closure of \mathbb{Z}[2i] in K.
    ( i=\sqrt{-1})
    It is easy to see that K=Q(i). Now, is i \notin \mathbb{Z}[2i] integral over \mathbb{Z}[2i]? Think about x^2-2ix-1 \in \mathbb{Z}[2i][x]. Can you see that the integral closure of \mathbb{Z}[2i] in K is simply \mathbb{Z}[i]? .
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