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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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Much as I like this attitude, I can't help but feel the lower bound on answer length is there for a reason, and would also prefer an answer that couldn't have been produced by a magic 8-ball :) –  Ben Millwood Jun 2 '12 at 17:58
    
@benmachine: If you think there's anything more to be usefully said, please post an answer saying it. –  Chris Eagle Jun 2 '12 at 18:11
    
I don't have time, but perhaps a proof that the reasoning is correct, or at least a sketch of one? –  Ben Millwood Jun 2 '12 at 18:13

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