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

Abstract

Let $p$ be a prime, $m$ be a positive integer ( $m\ge 1$, and $m\ge 2$ if $p=2$), and ${\chi }_n$ be a multiplicative complex character on ${\mathbb{F}}_{p^m}^{\ast }$ 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 ${\mathbb{F}}_{p^m}^{\ast }$ 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 $\mathbb{Q}[x_1,\dots ,x_n]/⟨x_1^p-1,\dots ,x_n^p-1⟩$.

arXiv comment10 pages, submitted