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Math Help - Ideals of the Gaussian Integers

  1. #1
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    Ideals of the Gaussian Integers

    Hello!

    Given the Gaussian Integers  Z[i] = \{ a + bi | a,b \in Z\}

    Then I am trying to show that

    1) I = \{ a + bi\ |\ a,b \in Z, 5|a, 5|b\} is an ideal in Z[i] but  I is not maximal. To be clear, I mean 5 divides a, and 5 divides b.

    To show it's an ideal, I have to show that it "absorbs" any element from Z[i] "into"
    it... seems like something that should follow from number theory. Also, I am not sure how it is not maximal! I need to find some other ideal that I is properly contained in? I can't think of any!

    2) J = <2+ i> is NOT a prime ideal in Z[i] (<> refers to the principal ideal)...

    and then I am trying to figure out how many elements there are in the factor ring \frac{Z[i]}{J}

    Any guidance or assistance appreciated! Thanks!!
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  2. #2
    Senior Member roninpro's Avatar
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    For part one, you have to show that for any element z=x+yi\in \mathbb{Z}[i], zI\subseteq I. Did you try multiplying (x+yi)(a+bi), and checking the condition?

    To finish that problem, try showing that I\subset \langle 1+i\rangle.

    For part two, if you can decide what \mathbb{Z}[i]/J is, then you can resolve your question about the primality of J. (Recall that an ideal I is a prime ideal if and only if R/I is an integral domain.)
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