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I would like to know why $\mathbb{C}[x,y]$ is not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings.

Thank you! 1

  • 0
    Do you have a specific map that you thought should be an isomorphism?2010-12-06
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    @user4465: Essentially, you only have bilinearity over $\mathbb{Z}$ in the latter.2010-12-06
  • 0
    No, Alex. I have no specific map.2010-12-06
  • 0
    The second one isn't even an integral domain. Note: $\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}$ has 4 square roots of 2.2010-12-06
  • 0
    ...in fact, there's infinitely many2010-12-06
  • 0
    Is there a simple definition of $\otimes _{\mathbb{Z}}$? Thanks2010-12-06
  • 0
    @Ross: http://en.wikipedia.org/wiki/Tensor_product_of_algebras2010-12-06

3 Answers 3