Polynomial whose only values are squares

Question

Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?

Answer 1

Yes, see this answer, which gives a proof, and references this article, which deals with the multivariate case.