On the Sum of Additive Characters and its Applications over Finite Fields

Abstract: In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified (\mathbb{F}_q)-Order. We establish a general formula for these character sums, providing an additive analogue to classical results previously known for multiplicative characters. As an application, we derive a M\"obius function (\mu(g)) for polynomials (g \in \mathbb{F}_q[x]), analogous to the integer M\"obius function (\mu(n)), and develop a characteristic function for (k)-normal elements. We also generalize several classical identities from the integer setting to the polynomial setting, highlighting the structural parallels between these two domains.