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

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 $\mathbb Q[x_1,\ldots,x_n]/\langle x_1^p-1, \ldots, x_n^p-1 \rangle$.