Suppose that $p(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$ is a monic irreducible polynomial over $\mathbb{Z}$. When does $p(x^k), k\geq1$ have irreducible factor $(x-\sqrt[k]{x_1})(x-\sqrt[k]{x_2})\cdots(x-\sqrt[k]{x_n})$?
For example $p(x)=x^3-6x^2+5x-1$ has roots $x_1=5.0492..., x_2=0.643104..., x_3=0.307979...$ and $p(x^2)=(x^3-2x^2-x+1)(x^3+2x^2-x-1)$ where $x^3-2x^2-x+1=(x-\sqrt[3]{x_1})(x-\sqrt[3]{x_2})(x-\sqrt[3]{x_3})$.