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Shift-invariant basis for polynomials over F_p

Source: Alibaba Global Math Competition 2021, Problem 17

July 4, 2021
algebrapolynomialnumber theorycollege contestsfinite field

Problem Statement

Let pp be a prime number and let Fp\mathbb{F}_p be the finite field with pp elements. Consider an automorphism τ\tau of the polynomial ring Fp[x]\mathbb{F}_p[x] given by τ(f)(x)=f(x+1).\tau(f)(x)=f(x+1). Let RR denote the subring of Fp[x]\mathbb{F}_p[x] consisting of those polynomials ff with τ(f)=f\tau(f)=f. Find a polynomial g∈Fp[x]g \in \mathbb{F}_p[x] such that Fp[x]\mathbb{F}_p[x] is a free module over RR with basis g,τ(g),…,τp−1(g)g,\tau(g),\dots,\tau^{p-1}(g).
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