Corollary 2

Let $\Omega$ be an extension of a field $K$ and $L, M$ two subextensions of $\Omega$. Suppose that $L$ is separable over $K$ and that the relative separable closure of $K$ in $M$ is equal to $K$. If $L$ and $M$ are algebraically disjoint over $K$, then they are linearly disjoint over $K$.

By Prop. 10 it is enough to remark that the ring $L \otimes_K M$ is an integral domain (V, p. 140, Cor.).

Exercises