REU 2006 · Discrete Math

Definition 9.1.3. An additive character of F is a character of the additive group F, i.e. a map χ : (F,+) → T, with χ(a+ b) = χ(a)χ(b). Now, let Fq denote the field of order q = p . We can define it by Fq := Fp[x]/(f), where f is any irreducible polynomial of degree k. We know that Fq is a cyclic group of order q − 1, and is generated by some g ∈ Fq (in other word Fq = 〈g〉). That is, g = 1 and no smaller positive power of g is 1. We have (χ(g)) = χ(g) = 1 for any multiplicative character g. So characters correspond to a choice of primitive (q − 1)-st root of unity ω, so that χ(g) = ω. Then, for any element x = g ∈ Fq , we have χ(x) = χ(g) = ω. In general, we have