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Let $F$ be a field, and $x_{1}, \dots, x_{n}$ free variables. Let $K=F(x_{1},x_{2},\dots,x_{n})$, being the fraction field of the ring of polynomials in the variables $x_{1},\dots,x_{n}$. Show that $S_{n} \subset \mathrm{Aut}(K)$, and in particular, use this result to show that any finite group $G$ is the Galois group of an extension of fields.


I'm lost on how to show this. I've seen other answers around, but they invoke the Fundamental Theorem of Galois Theory (which I haven't seen). What would be a way to get started on this problem? (and if possible without using the Fundamental Theorem of Galois Theory). I know that for the second statement, one would need to use Cayley's Theorem at some point.

Thanks for the help!

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