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How many solutions?

Source: 2016 Korea Winter Program Test1 Day1 #4

January 27, 2016
polynomialnumber theorynumber theory proposedalgebra

Problem Statement

p(x)p(x) is an irreducible polynomial with integer coefficients, and qq is a fixed prime number. Let ana_n be a number of solutions of the equation p(x)≡0mod  qnp(x)\equiv 0\mod q^n.
Prove that we can find MM such that {an}n≥M\{a_n\}_{n\ge M} is constant.
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