Corollary 1

*Let $K, E, F$ be three fields such that $K \subset E \subset F$ and $[F : K]$ is finite. Then the degrees $[E : K]$ and $[F : E]$ are divisors of $[F : K]$.

If the degree $[F : K]$ is prime, there is thus no subextension of $F$ other than $K$ and $F$. But note that when $[F : K]$ is not prime, there is not necessarily a subextension of $F$ other than $K$ and $F$ (cf. V, p. 146, Exercise 1).*