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

TL;DR

This paper investigates additive character sums over finite fields, deriving formulas and applications such as a polynomial M"obius function and characteristic functions for normal elements, extending classical identities to polynomial rings.

Contribution

It introduces a general formula for additive character sums over finite fields and applies it to define a polynomial M"obius function and characteristic functions for k-normal elements, bridging integer and polynomial identities.

Findings

Derived a general formula for additive character sums over finite fields.

Defined a M"obius function for polynomials analogous to the integer case.

Extended classical identities from integers to polynomials.

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.