abstract algebra - Tensor product and compositum of linearly disjoint field extensions

Let $k$ be a field and $K/k$ and $L/k$ be two field extensions that are linearly disjoint over $k$. If $K$ and $L$ are finite extensions then it easily follows that the compositum $KL$ and the tensor product $K\bigotimes_k L$ are isomorphic as fields.

My question, is this still true when $K$ and $L$ are infinite extensions?

Edit: It seems unlikely if neither $K$ nor $L$ is algebraic as pointed out below by Jyrki Lahtonen. So, let us assume that $L$ is algebraic over $k$.

Answer

With the assumption that $L/k$ is algebraic this is indeed correct. An outline of an argument is as follows:

Since $L$ is an algebraic extension without loss of generality we may assume that it is finite (since an algebraic extension is a union of its finite subextensions). It is clear (at least to me) that $K\bigotimes_k L$ is a domain (this is true even if $L/k$ is not algebraic).

Let $\{l_1,\ldots,l_n\}$ be a basis of $L/k$ then $1\otimes l_1,\ldots, 1\otimes l_n$ is a basis for $K\bigotimes_k L$ over $K$. Therefore $K\bigotimes_k L$ is a finitely generalted $K$ module which is also a domain. Then it must indeed be a field.