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Math Help - Show that a prime is not irreducible in Z[i]

  1. #1
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    Show that a prime is not irreducible in Z[i]

    Let p be prime with  p \equiv 1 (mod \ 4) , show that p is not irreducible in Z[i].

    Proof. Now from the previous problem, I know that p \ n^2 + 1 = (n+i)(n-i).

    If p is irreducible, then p is a prime and it divides either (n+i) or (n-i). I'm trying to show that is impossible, any hint?

    Thank you.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let p be prime with  p \equiv 1 (mod \ 4) , show that p is not irreducible in Z[i].

    Proof. Now from the previous problem, I know that p \ n^2 + 1 = (n+i)(n-i).

    If p is irreducible, then p is a prime and it divides either (n+i) or (n-i). I'm trying to show that is impossible, any hint?

    Thank you.
    By your other problem if p\equiv 1 (\bmod 4) then p|(n^2+1) for some n\in \mathbb{Z}. Thus, p|(n+i)(n-i), assume that p is irreducible then p|(n+i) or p|(n-i). If p|(n+i) then n+i = p(a+bi) by definition for some a,b\in \mathbb{Z}. Thus, n+i = pa+pbi \implies 1 = pb which is an impossibility. Similarly, p cannot divide n-i. Thus, the assumption that p is irreducible is false.
    Last edited by ThePerfectHacker; January 21st 2008 at 02:18 PM.
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