Let $z\in \mathbb{C}$ be an algebraic integer, and let $P\in \mathbb{Z}[X]$ be its minimal polynomial over $\mathbb{Z}$. Prove that $(X-z)^2 \nmid P(X)$.
My attempt: My first idea was to prove that $P'(z)\neq0$ because $z$ would be a root of the derivative of $P$ if and only if it would be a repeated root, but I don't know anything about the structure of $P'$ so I this was the end for this idea. On an intuitive level I understand why this statement should be true because minimal polynomial basically means "what other factors are we forced to add to $(X-z)$ to make it an integer polynomial", and it seems logical that it makes no sense to add the same factor again. However, I am unable to formalize this idea.