Let $f(x), g(x)$ be irreducible polynomials over $\mathbb{Q}$, is $f(g(x))$ also irreducible over $\mathbb{Q}$

Question

Let $f(x), g(x)$ be irreducible polynomials over $\mathbb{Q}$, is $f(g(x))$ also irreducible over $\mathbb{Q}$?

I have no idea how to prove it or give a counterexample. Appreciate any help!

Answer 1

Let $f(x)=x^2+16$ and $g(x)=x^2+3.$

Thus, $$f(g(x))=(x^2+3)^2+16=x^4 +6x^2+25=$$ $$=(x^2+5)^2-4x^2=(x^2-2x+5)(x^2+2x+5).$$