关于$\mathbb F^*_{p^m}$及相关多项式代数的纤维特征
On the Fiber Characters of $\mathbb F^*_{p^m}$ and related Polynomial Algebras
摘要：
设$p$是素数，$m$是正整数（$m \geq 1$，如果$p=2$则是$m \geq 2$），$χ_n$是$\mathbb F^*_{p^m}$上的乘法复数字符，阶为$n| (p^m-1)$。我们证明了$\mathbb F^{*}_{p^m}$的一个分区$\mathcal A_1 \cup \mathcal A_2 \cup \cdots \cup \mathcal A_n$是$χ_n$的纤维分区当且仅当这些纤维% $\mathcal A_i$满足某些可加性.这相当于表明，这些纤维的多元特征多项式的集合，完成与常数多项式$1$，是一个$(n+1)$维交换代数的基础与单位元环$\mathbb Q[x_1,\ldots,x_n]/\langle x_1^p-1, \ldots, x_n^p-1 \rangle$。
英文摘要：
Let $p$ be a prime, $m$ be a positive integer ( $m \geq 1$, and $m \geq 2$ if $p=2$), and $χ_n$ 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 $χ_n$ 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 $\mathbb Q[x_1,\ldots,x_n]/\langle x_1^p-1, \ldots, x_n^p-1 \rangle$.
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