Here are some of my thoughts.
- Use group theory to show contradiction.
Suppose K is a split field of an irreducible polynomial $f$, $deg f=m>n$. Then $S_n$ isomorphic to a transitive subgroup of $S_m$. It's sufficient to show that $S_n$ cannot embed into $S_m$ as a transitive subgroup. However, I can't get it.
- Use primitive element theorem to construct a $n$ degree irreducible polynomial.
$S_{n-1}\subseteq S_n=Gal(K/F)$ Let $\alpha$ be the primitive element(i.e. $K=F(\alpha)$), let $$\beta =\sum_{\sigma \in S_{n-1}}\sigma \alpha$$ Let $O$ be the orbit of $\beta$ under $Gal(K/F)$.$$f(X)=\Pi_{u\in O}(X-u)$$ Then $f(X)$ is $Gal(K/F)$-invariant hence $f(X)\in F[X]$
It's sufficient to show $|O|=n$, but I'm stuck in this last step.