Isomorphism from Z7[x] to Z

  • #1

Homework Statement



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


Homework Equations





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.
 

Answers and Replies

  • #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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