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Let $f$ be an irreducible polynomial of degree 4 over $\Bbb Q$ and $Gal(f)=S_4$.

Prove that there isn't nontrivial intermediate field between $\Bbb Q(\alpha)$ and $\Bbb Q$ where $\alpha$ is a root of $f$.

Here is my attempt:

Let $E$ be a splitting field of $f$. Then, $[E:\Bbb Q]=24$.

If there is such intermediate field $K$, then clearly $[E:K]=12$, which means $Gal(E/K)=A_4$.

But it's impossible? Or should I do different way by using irreducibility of $f$?

Arturo
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    You are well on your way. +1 for that and explaining your thinking! The catch is that $Gal(E/\Bbb{Q}(\alpha))= S_3$, i.e. a point stabilizer inside $S_4$. But $A_4$ does not contain any of the four point stabilizers. See the question I proffer as a duplicate for details (and a generalization). – Jyrki Lahtonen Dec 11 '13 at 13:06
  • @JyrkiLahtonen Oh, thanks very much! – Arturo Dec 11 '13 at 13:08
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    @JyrkiLahtonen Yeah, I had to think about $Gal(E/\Bbb Q(\alpha))$.... I should bear in mind :) – Arturo Dec 11 '13 at 13:12

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