I have two questions about a construction introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122):
We fix an integral proper normal curve $X$ over a field $k$. We consider it's function field $K$ which is a finite extension of $k(t)$ and take an arbitrary field extension $L \vert k$.
The point of interest is the resulting tensor product $K \otimes L$. We know that $K \otimes L$ is finite dimensional $L(t)$-algebra.
Following two questions:
Assume $K \otimes L$ is a finite direct product of fields $L_i$. Why these fields are finitely generated (as $L$-modules)?
Assume non $k$ is algebraically closed. Why is $K \otimes L$ then a field?