Question. For a monic, irreducible cubic polynomial $p(x)\in \Bbb Z[x]$ with three real roots, prove or disprove that $p(x^3)$ must be irreducible over $\Bbb Z$ as well.
Here the notation $p(x^3)$ means substituting $x\mapsto x^3$ so we get a degree $9$ monic polynomial. I failed to find counterexamples.
My attempt: Let $\omega$ be a primitive $3^{rd}$ root of unity over $\Bbb Q$ then if the roots of $p(x)$ are $a^3,b^3,c^3$ then the roots of $p(x^3)$ are $a,\omega a, \omega^2a,b,\omega b, \omega^2b,c,\omega c, \omega^2c$.