# Galois extension corresponding to $S_n$

Given a positive integer $n$. How to find a Galois extension $K/F$ such that $Gal(K/F)=S_n$?

With the restriction $F=\mathbb{Q}$, this is the Inverse Galois Problem. But if we are allowed to choose $F$ and $K$, I suspect there is a simpler example.

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How about $F=\mathbb Q(T_1, \ldots, T_{n-1})$ and $K=\mathbb F[X]/(X^n+T_{n-1}X^{n-1}+\ldots +T_0)$?
Why is $K$ Galois over $F$? Maybe the construction needs to be done several times to get a Galois extension. Or simply take $K$ to be the splitting field of $X^n+T_{n-1}X^{n-1}+\cdots+T_0$ over $F$? –  user83058 Jun 19 '13 at 6:02