Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 18H / i / Solution

TakeIt is a nonzero polynomial, but every element of the finite field ${\mathbb{F}}_q$ satisfies ${\alpha }^q=\alpha$. Thus it vanishes on every point of ${\mathbb{F}}_q^n$. Formal nonzero polynomials and nonzero polynomial functions need not coincide over a finite field.