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1999 Romania National Olympiad, 4

Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty. Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$

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