Can the Galois group of $x^5-5x^3+4x+1 \in \mathbb{Q}[x] $ be isomorphic to $S_5$?
I know that it has five real roots, then I think that it is impossible. I think that this Galois group cannot contain a 2-cycle. Any idea about how I can prove this?
2026-03-30 12:28:14.1774873694
Galois group of $x^5-5x^3+4x+1 \in \mathbb{Q}[x] $
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