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Just a quick sanity check. Am I right in thinking that the Galois group of $X^4+X^3+1$ over $\mathbb{Q}$ is isomorphic to $S_4$? It's irreducible (since it's irreducible mod 2) and so strictly separable. Hence reducing (mod p) tells us what cycles the Galois group has. But reducing (mod 2) shows it must contain a 4 cycle and reducing (mod 3) tells us it must contain a 3 cycle. But the only transitive subgroup of $S_4$ with these properties is $S_4$ itself. Is this reasoning correct?

Many thanks!

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Yes.${}{}{}{}{}{}{}{}{}{}{}{}$

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