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Suppose $K/k$ is a finite Galois extension such that the Galois group of $K/k$ is isomorphic to $\mathcal S_4$. How can we show that $K$ is a splitting field for some degree $4$ polynomial $f(x)$ in $k[x]$?

I first thought to try that since $K/k$ is finite and separable, it is simple. We can the realize $K$ as the splitting field of the minimal polynomial of a primitive element. In particular, the set of irreducible polynomials $f(x)$ in $k[x]$ such that $K$ is a splitting field of $f(x)$ is non-empty.

So we can take $f(x)$ in this set of minimal degree. If I can show that the degree of $f(x)$ must be $4$ I am done. So far I can show that if $\deg(f)$ is at least $4$, then $\deg (f)$ must divide $24$, since the Galois group of $K/k$ acts transitively, on the set of roots of $f(x)$ in $K$.

But I have not gotten any further with this approach. Am I heading in a wrong direction?

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  • $\begingroup$ Maybe applying Galois theory? Finding a nonnormal subgroup of index $4$? $\endgroup$ – Lubin Apr 2 '15 at 2:55

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