Rationality

Abstract

Multiplicative character theory provides a natural setting for developing the complexity results of Auslander-Feig- Winograd [1]. The first reason for this is the simplicity of the formulas describing the action of FT on multiplicative characters. We will now discuss a second important property of multiplicative characters. In a sense defined below, the spaces spanned by the subsets V _k , 1 ≤ k > m, of multiplicative characters are rational subspaces. As a consequence, we will be able to rationally manipulate the FT matrix F(p ^m) into block diagonal matrices where each block action corresponds to some polynomial multiplication modulo a rational polynomial of a special kind. This is the main result in the work of Auslander-Feig-Winograd. Details from the point of view of multiplicative character theory can be found in [2].