Suppose $K$ is a splitting field over $F$ such that $[K:F]=n$. Prove that $K$ is a splitting field over $F$ for any irreducible polynomial of degree $n$ of $F(x)$ having a root in $K$.
Well, let the polynomial be $p(x)$, and the root be $c$ (so $p(c)=0$).
Consider $F(c)$. We know that $[F(c):F]=n$, and we want to show that $K=F(c)$.
Since $c\in K$, we have $F(c)\subseteq K$. We're left to show $K\subseteq F(c)$.
How do we show that?