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Math Help - Factorization of x in Z_4[x]

  1. #1
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    Factorization of x in Z_4[x]

    I was hoping someone could give me a hand with this little problem that's been driving me a bit nuts:

    Show that there is a factorization x = f(x)g(x) in Z_4[x] (the polynomial ring over the integers mod 4) such that neither f(x) nor g(x) is a constant.

    Is there some kind of method or good way to find these factors, or is it just a guess and check situation? I've been searching for these factors for a while with no luck (haven't strayed beyond quadratics yet).
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mtdim View Post
    Is there some kind of method or good way to find these factors, or is it just a guess and check situation? I've been searching for these factors for a while with no luck (haven't strayed beyond quadratics yet).

    Taking into account that

    2\cdot 2=0,\;3\cdot 3=1

    we obtain

    (2x+3)^2=0x^2+0x+1=1

    So, choose

    f(x)=x(2x+3),\;g(x)=2x+3


    Fernando Revilla
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