Maithri K.

arXiv2026

On $\mathbb{F}_q$-Order of Polynomials and Properties of $r$-Primitive and $k$-Normal Elements over Finite Fields

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been extensively studied. In this paper, we investigate several properties of $r$-primitive and $k$-normal elements. Furthermore, by using the concept of the $\mathbb{F}_q$-Order of a polynomial, we explore properties of $k$-normal polynomials.

math.RA↗

arXiv2025

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.

math.NT↗

On \(\mathbb{F}_q\)-Order of Polynomials and Properties of \(r\)-Primitive and \(k\)-Normal Elements over Finite Fields

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