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Prove that there exist polynomials A0,A1,...,Ak

Source: Iran TST 2011 - Day 3 - Problem 2

May 14, 2011
algebrapolynomialinductionmodular arithmeticnumber theory

Problem Statement

Let pp be a prime and kk a positive integer such that k≤pk \le p. We know that f(x)f(x) is a polynomial in Z[x]\mathbb Z[x] such that for all x∈Zx \in \mathbb{Z} we have pk∣f(x)p^k | f(x).
(a) Prove that there exist polynomials A0(x),…,Ak(x)A_0(x),\ldots,A_k(x) all in Z[x]\mathbb Z[x] such that f(x)=∑i=0k(xp−x)ipk−iAi(x), f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),
(b) Find a counter example for each prime pp and each k>pk > p.
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