On the Fiber Characters of $\mathbb F^*_{p^m}$ and related Polynomial Algebras

TL;DR

This paper characterizes the fiber partitions induced by multiplicative characters on finite fields and relates them to a basis of a specific algebra of multivariate polynomials, revealing algebraic structures underlying these partitions.

Contribution

It provides a characterization of fiber partitions of finite fields via multiplicative characters and links these to a basis in a polynomial algebra, extending understanding of algebraic structures in finite fields.

Findings

Fiber partitions correspond to additive properties of characteristic polynomials.

The set of characteristic polynomials forms a basis of a specific commutative algebra.

The algebra has dimension n+1 with an explicit polynomial basis.

Abstract

Let $p$ be a prime, $m$ be a positive integer ( $m \geq 1$, and $m \geq 2$ if $p=2$), and $\chi_n$ be a multiplicative complex character on $\mathbb F^*_{p^m}$ with order $n| (p^m-1)$. We show that a partition $\mathcal A_1 \cup \mathcal A_2 \cup \cdots \cup \mathcal A_n$ of $\mathbb F^{*}_{p^m}$ is the partition by fibers of $\chi_n$ if and only if these fibers %$\mathcal A_i$ satisfy certain additive properties. This is equivalent to show that the set of multivariate characteristic polynomials of these fibers, completed with the constant polynomial $1$, is the basis of a $(n+1)$-dimensional commutative algebra with identity in the ring $\mathbb Q[x_1,\ldots,x_n]/\langle x_1^p-1, \ldots, x_n^p-1 \rangle$.