$f(x)$ is irreducible but $f(x^n)$ is reducible

Question

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible?

Answer 1

No. If $f(X)=g(X)h(X)$, then $f(X^n)=g(X^n)h(X^n)$.