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Number of Polynomial Q such that P(x) | P(Q(x))

Source: IZHO 2021 P6

January 9, 2021
algebrapolynomial

Problem Statement

Let P(x)P(x) be a nonconstant polynomial of degree nn with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials Q(x)Q(x) of degree less than nn with rational coefficients such that P(x)P(x) divides P(Q(x))P(Q(x)) a) is finite b) does not exceed nn.
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