TITLE: Why is $\mathbb{C}[x,y]$ not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings?
QUESTION [9 upvotes]: 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

REPLY [9 votes]: To show that $1\otimes 1+i\otimes i$ is nonzero
in $\mathbb{C}[X]\otimes_{\mathbb{Z}}\mathbb{C}[X]$ note that it maps
to $1\otimes 1+i\otimes i$ in
$\mathbb{C}[X]\otimes_{\mathbb{R}}\mathbb{C}[X]$.
This is a tensor product over a field, so a basis as an
$\mathbb{R}$-vector space is got by tensoring togther bases
on each side. Now as the set of elements of the forms $X^n$
and $iX^n$ are bases of $\mathbb{C}[X]$ over $\mathbb{R}$
then $1\otimes 1$ and $i\otimes i$ are linearly independent
over $\mathbb{R}$.
As
$$(1\otimes 1+i\otimes i)(1\otimes 1-i\otimes i)=0$$
then $\mathbb{C}[X]\otimes_{\mathbb{Z}}\mathbb{C}[X]$
isn't an integral domain, unlike $\mathbb{C}[X]\otimes_{\mathbb{C}}\mathbb{C}[X]$.