I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof.
It states like this:
Let $K$ be a field and $f,g \in K[x]$. Let $\alpha$ be a root of $f$ in the algebraic closure of $K$. Then $f \circ g$ is irreducible over $K$ if and only if $f$ is irreducible over $K$ and $g-\alpha$ is irreducible over $K(\alpha)$.
Can you please give a proof for this?