# Galois group is $S_n$ then $L$ is the splitting field of a degree $n$ polynomial

I am revising Galois Theory and am faced with the following problem. Let $L/K$ be a finite Galois extension whose Galois group is isomorphic to $S_n$. Show that L is the splitting field of a separable polynomial of degree $n$. My idea, is that since the extension is normal, we know $L =$ Split$_K(f)$ for some $f\in K[X]$, so we want to use Orbit-Stabilizer or something to show $deg(f) = n$.

• "Revising" or "Revisiting"? ;-) – Adrian Keister Apr 26 '18 at 14:41
• Sorry? I Haven't seen this problem before. – Elie Bergman Apr 26 '18 at 14:45
• If you are "revising" Galois Theory, you are changing the theory, which, considering how well-established it is, I wouldn't advise you do. If you are "revisiting" Galois Theory, then you are re-learning it, which is a great thing to do. – Adrian Keister Apr 26 '18 at 14:48
• – Eric Towers Apr 26 '18 at 14:48