Are $\mathbb{C}(x)\otimes_\mathbb{R}\mathbb{C}(y)$ and $\mathbb{C}(x)\otimes_\mathbb{C}\mathbb{C}(y)$ integral domains or fields?

Question

Question

Let $\mathbb{C}(x)$ and $\mathbb{C}(y)$ be function fields of one variable. Consider $\mathbb{C}(x)\otimes_\mathbb{R}\mathbb{C}(y)$ and $\mathbb{C}(x)\otimes_\mathbb{C}\mathbb{C}(y)$.

(a) Determine if they are integral domains.

(b) Determine if they are fields.

Answer

For the latter tensor product, I am pretty sure that we can write the following isomorphism

\begin{align*} \mathbb{C}(x)\otimes_\mathbb{C}\mathbb{C}(y)&\to \mathbb{C}(x,y)\\ x\otimes1&\mapsto x\\ 1\otimes y&\mapsto y \end{align*}

This shows that $\mathbb{C}(x)\otimes_\mathbb{C}\mathbb{C}(y)$ is both an integral domain and a field. However, I don't know what to do with the former tensor product. Any help/hint would be appreciated. Thanks in advance...