Additive character : For any field or for a finite field?

Question

We can define an additive character for any field, can't we? The reason why i'm asking this question is that when i google "additive character", all definitions i have seen are for a finite field. If we can define an additive character, could you give a link to the definition of an additive character for any field?

Answer 1

A quick google search might have found a text about the Finite Fields Waring Problem which - in spite of its title - on page 3 gives the very general

Definition 2.1. Let $G$ be an abelian group. A character of $G$ is a homomorphism from $G$ to $\mathbb C^\times$. A character is trivial if it is identically $1$. We denote the trivial character by $\chi_0$ or $\psi_0$.