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

Let p be a prime, m be a positive integer ( m ≥ 1, and m ≥ 2 if p = 2), and χn be a multiplicative complex character on F∗ p with order n|(p−1). We show that a partition A1∪A2∪· · ·∪An of F∗pm is the partition by fibers of χn if and only if these fibers satisfy certain additive properties. This is equivalent to showing that the set of multivariate characteristic polynomials of these fibers, completed with the constant polynomial 1, is the basis of an (n + 1)-dimensional commutative algebra with identity in the ring Q[x1, . . . , xn]/〈x p 1 − 1, . . . , x n − 1〉. Mathematics Subject Classification (2000): 11A15, 11N69, 11R32