Corollary 2

*Let $K, E$ and $F$ be three fields such that $K \subset E \subset F$. Suppose that $[F : K]$ is finite, then the relation $[E : K] = [F : K]$ is equivalent to $E = F$ and the relation $[F : E] = [F : K]$ is equivalent to $E = K$.

For if $L$ is an extension field of $L'$ then $[L : L'] = 1$ is equivalent to $L' = L$.*