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

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.