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Functional equation on functions on F_p

Source: Miklos Schweitzer 2020, Problem 8

December 1, 2020
algebrafunctional equationnumber theoryfunction

Problem Statement

Let Fp\mathbb{F}_{p} denote the pp-element field for a prime p>3p>3 and let SS be the set of functions from Fp\mathbb{F}_{p} to Fp\mathbb{F}_{p}. Find all functions D ⁣:S→SD\colon S\to S satisfying D(f∘g)=(D(f)∘g)⋅D(g)D(f\circ g)=(D(f)\circ g)\cdot D(g) for all f,g∈Sf,g \in {S}. Here, ∘\circ denotes the function composition, so (f∘g)(x)(f\circ g)(x) is the function f(g(x))f(g(x)), and ⋅\cdot denotes multiplication, so (f⋅g)=f(x)g(x)(f\cdot g)=f(x)g(x).
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