Let $K \leq L$ be a finite Galois extension whose Galois group is isomorphic to $S_n$. I want to show that $L$ is the splitting field of some polynomial of degree $n$ over K.
So far, I thought of picking $\alpha \in L$, letting $\alpha_1, \dots , \alpha_m$ be the orbit of $\alpha$ under the action of Gal$(L/K)$ (as in the proof of Artin's Lemma) and considering the polynomial:
$$ f(t) = \prod_{i=1}^m (t - \alpha_i)$$
I want to say, pick $\alpha \in L $ minimising $m$ ($\alpha \notin K$), and show then that $m = n$, in which case $L$ is the splitting field of $f$ over $K$. I'm not sure whether this is the right approach from here, or how to proceed. Any help or hints would be greatly appreciated. Thanks!