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