If $f(x)$ is irreducible, is $f(x^k)$ irreducible? Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer?

For $\alpha=\sqrt[6]{\sqrt{2}+\sqrt{3}}$, I saw that it satisfies a polynomial over $\mathbb{Q}$ given by $x^{24}-10x^{12}+1$. I wanted to check whether this polynomial is irreducible, and I came to above general question.

If $k\ne2$ then
$f(x)=x^2-2^k$ is irreducible but
$f(x^k)=(x^2)^k-2^k$ is reducible
