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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]$?

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

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