## Field extensions of prime degree have no intermediate subfields

Let $K/F$ be a field extension with prime degree $p$. Show that any subfield of $K$ containing $F$ is either $K$ or $F$.

Let $F \subseteq E \subseteq K$. Then $[K:E][E:F] = [K:F] = p$ by Theorem 14 in D&F. Since the degree of a (finite) field extension is an integer, either $[K:E] = 1$ (so $E = K$) or $[E:F] = 1$ (so $E = F$).