I aim to show that the roots of $x^4-4x+2$ is not constructible by ruler and compass. Which is left to prove is that the splitting field is exactly $S_4$. But I have no idea to prove it so far… May I please ask some possible method to do it? Thanks!
2026-03-28 22:29:52.1774736992
Prove the Galois group of the splitting field of $x^4-4x+2$ is $S_4$
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Factor modulo $p$ for various primes. If the group really is $S_4$ your polynomial $f$ will factor modulo $p$ in all possible ways. If you find a $p_1$ with $f$ irreducible modulo $p_1$ and a $p_2$ with $f$ having an irreducible cubic factor modulo $p_2$, then the Galois group will have elements of cycle structures $(4)$ and $(3\,1)$. The only possibility will then be $S_4$.