Splitting Fields - Field Extensions

Question

How do I demonstrate this exercise?

Let $K$ be a splitting field of $f(x)$ over $F$. If $[K:F]$ is prime, $u \in K$ is a root of $f(x)$, and $u \notin F$, show that $K = F(u)$.

Thanks for any comments or help

Answer 1

Note that $[K:F]=[K:F(u)][F(u):F]$ is prime, so, for any $u \in K$, either $[F(u):F]=1$ ie $u \in F$, or $[K:F(u)]=1$ ie $K=F(u)$.