On Fiber Characters of and Related Polynomial Algebras

Abstract

Let p be prime, m a positive integer( (m ≥ 1, and m ≥ 2 if p = 2), and χ_n a multiplicative complex character on with order n | (p^m = 1). We show that a partition of is the partition by fibers of χ_n 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 .

Subject Classification: (2010):

• 11A15
• 11N69
• 11T22
• 12E20
• 13F20

Keywords:

• nth power residue
• Cyclotomy
• Dirichlet characters
• Finite fields
• Polynomial rings