Let $\mathbb{C}$ and $R$ be complex and real number fields. Let $C(x)$ and $C(y)$ be function fields of one variable. Consider $\mathbb{C}(x) ⊗_R \mathbb{C}(y)$ and $\mathbb{C}(x) ⊗_\mathbb{C} \mathbb{C}(y)$.
(1). Determine if they are integral domains.
(2). Determine if they are fields.
I have learned finite seperable field extension tensor product over field. But clearly they are both infinite dimensional. And for the second tensor, I know they are linearly disjoint over $\mathbb{C}$ but $x,y$ is not algebraic over $\mathbb{C}$
For $\mathbb{C}(x) ⊗_R \mathbb{C}(y)$, I don't have any idea. Maybe let $\mathbb{C}(x)=\mathbb{R}(i,x)$? But how to show its structure?