Suppose that $f$ is a polynomial in $Z[X]$, such that $f = (X-\alpha_1) ... (X-\alpha_n)$ with $n$ distinct complex, irrational roots. Suppose that $Q[\alpha_i]$ = $Q[\alpha_1, ..., \alpha_n]$ for all $i$. Question: Does this imply that $f$ is irreducible?
2026-04-04 13:40:38.1775310038
Is this condition enough for irreducibility?
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Counterexample: $f(x)=(x^2-2)(x^2-8)$.