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Math Help - Gaussian integers.

  1. #1
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    Gaussian integers.

    How can I prove that there is no isomorphism from Z[i]/<a+bi> to Zn, for any n, if (a,b) ≠1?
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    Quote Originally Posted by Deveno View Post
    How can I prove that there is no isomorphism from Z[i]/<a+bi> to Zn, for any n, if (a,b) ≠1?

    If by Zn you meant \mathbb{Z}_n:=\mathbb{Z}/n\mathbb{Z} , then you can't since \mathbb{Z}[i]/<a+bi>\cong \mathbb{Z}_{a^2+b^2}

    Tonio
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    is this true even if (a,b) ≠ 1? if (a,b) = 1, yes, i know that it's isomorphic to Z/(a^2+b^2)Z, although i would like to see a cleaner version of the isomorphism than what i came up with.

    but what if a+bi = d(k+mi)? for example, what is Z[i]/<4+6i>? really needing some help, here...
    Last edited by Deveno; April 5th 2011 at 09:53 PM.
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    Quote Originally Posted by Deveno View Post
    is this true even if (a,b) ≠ 1? if (a,b) = 1, yes, i know that it's isomorphic to Z/(a^2+b^2)Z, although i would like to see a cleaner version of the isomorphism than what i came up with.

    but what if a+bi = d(k+mi)? for example, what is Z[i]/<4+6i>? really needing some help, here...

    Ok, I missed that coprimality condition, but the answer's still the same, though it must be fixed.

    You really want to read http://home.wlu.edu/~dresdeng/papers/factorrings.pdf , in particular

    corollary 3.

    Tonio
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  5. #5
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    thanks, that is helpful.
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