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Isomorphism from Z7[x] to Z

  1. Nov 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Let R = Z7[x]. Show that R is not isomorphic to Z.


    2. Relevant equations



    3. The attempt at a solution

    One of the necessary conditions for an isomorphism f is that f be one to one. So consider 8x in Z. f(8x) = x, f(1x) = x. So f cannot be an isomorphism. I'm clearly missing something though, since this seems a bit too easy.
     
  2. jcsd
  3. Nov 7, 2013 #2
    Actually nevermind, it is that easy. There's a hint hidden that basically says "an isomorphism has to be one to one..."
     
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