Rational function field of product affine varieties Let $X, Y$ be affine varieties, we know that the coordinate ring of the product variety $X\times Y$ satisfies $k[X\times Y]\cong k[X]\otimes_k k[Y]$.
My question is is it true that for rational function field, we also have  $k(X\times Y)\cong k(X)\otimes_k k(Y)$? If not, is there a way to relate $k(X\times Y)$ with $k(X)$, and $k(Y)$ ?

It is not true in general that $k(X\times Y)\cong k(X)\otimes_k k(Y)$. 
For example, let $k=\mathbb{Q}$, $k[X]=k[Y]=\mathbb{Q}(i)$ then $k(X)=k(Y) = \mathbb{Q}(i)$, but $\mathbb{Q}(i) \otimes_{\mathbb{Q}} \mathbb{Q}(i)$ is not even a field.
It is also not true for product of two affine lines: $k(x) \otimes_k k(y) \subsetneq k(x,y)$.
