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If $K/F$ is Galois extension with Galois group $S_{n}$ then show that $K$ is the splitting field of a degree $n$ polynomial irreducible over $F$. We know $K$ is splitting field of some separable polynomial $f$. If $f$ has $m$ roots (so degree of $f$ is $m$) then $Gal(K/F)$ sits inside $S_{m}$, so $m\geq n$. If $m=n$ then I can show $f$ is irreducible. But how to show $m=n$?

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The Galois group acts transitively on the zeros of the polynomial $f$. Under $S_n$, each zero can have at most $n$ images, hence, at most $n$ conjugates, hence the degree of the polynomial can be at most $n$.

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