A corollary on page 287 of Hungerford's Algebra is
Corollary 6.13. If $F$ is an extension field of $E$ and $E$ is an extension field of $K$, then $$[F:E]_s[E:K]_s=[F:K]_s\mbox{ and } [F:E]_i[E:K]_i=[F:K]_i.$$
Here $[F:K]_s$ means the separable degree of $F$ over $K$ and $[F:K]_i$ the inseparable degree of $F$ over $K$.
This may be true, but the proof Hungerford gives seems to work only when $F$ is finite-dimensional over $K$. Am I missing something or does Hungerford drop an assumption?