Let $f(x),g(x)$ be polynomials in $\mathbb{Q}[x]$. If $\mathrm{deg}(f)\geq2$ and $f$ irreducible, is the composition $f(g(x))$ always reduced (has no repeated irreducible factors)?


(If we do not ask $\mathrm{deg}(f)\geq2$ we can take $f(x)=x-1, g(x)=x^2+1$; if we do not ask $f$ be reducible, we can take $f(x)=(x-1)x$ and $g=x^2+1$.)



Does irreducible polynomial remain reduced by pre-composition?
It may have repeated irreducible factor. Take $f(x)=x^2+1$ and $g(x)=x+f(x) h(x)$ so that $g'(i)=0$. Then $f^2$ divides $f(g) $.


