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Suppose $f\in \mathbb{Z}[x]$ is an irreducible polynomial with Galois Group $S_4$. If $\theta$ is a root of $f$. Say $K=\mathbb{Q}(\theta)$

  1. Prove that there is no field poperly contained in $\mathbb{Q}(\theta)/\mathbb{Q}$
  2. Is $\mathbb{Q}(\theta)/\mathbb{Q}$ a Galois Extension?
  3. Is $\theta$ constructible by ruler and compass?

If $E$ is the splitting field of $f$ over $\mathbb{Q}$. And since we are working over a field of characteristic $0$ and with an irreducible polynomial. So I will use the Fundamental Theorem of Galois Theory and using that I shall decide things about $\mathbb{Q}(\theta)$ I believe that $[\mathbb{Q}(\theta):\mathbb{Q}]=4$ and hence the corresponding Galois Group $H=\operatorname{Gal}(E/K)$ has order $6$

Now the only possibilites for $H$ are $\mathbb{Z}_6$ and $S_3$. Now I feel $H=S_3$ and if there are intermediate fields between $\mathbb{Q}(\theta)$ and $\mathbb{Q}$, then it gives me a subgroup of $S_4$ of order $12$ containing $S_3$. Which is a contradiction.

Am I correct? Can any help me with the solution because I am trying to understand things.

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    $S_4$ has no element of order $6$ so $H$ must be $S_3$. – ancient mathematician Jun 26 '22 at 06:40
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    Instead of looking for possible subgroups of order six it is possibly simpler to just observe that the full Galois group $S_4$ permutes the roots $\theta_i, i=1,2,3,4$, in all possible ways. Any automorphism from the subgroup $H$ must keep $\theta=\theta_1$ fixed, so $H$ is the point stabilizer of the natural action of $S_4$.That is, $H\simeq S_3$. You have the correct argument, $H$ is a maximal subgroup of $S_4$, meaning that there cannot be any intermediate fields between $\Bbb{Q}$ and $K$. – Jyrki Lahtonen Jun 28 '22 at 08:13
  • The reason why I posted that as a comment as opposed to as an answer is that I answered a more general version of this question here. I looked at quartic examples involving a similar argument here. – Jyrki Lahtonen Jun 28 '22 at 08:17
  • Anyway, because I answered those two variants, and I have two applicable dupehammers, I won't initiate the process of closing this as a duplicate. A) it's not kosher to redirect viewers to your own old posts by dupehammering (less violent methods are fine), B) it is always possible that I overlooked some nuances. Furthermore, we can afford to proceed slowly. I mean questions on Galois theory usually won't get a storm of quick answers here :-) – Jyrki Lahtonen Jun 28 '22 at 08:20

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