A Generalized Counting and Factoring Method for Polynomials over Finite Fields

摘要

We discuss a transform on the set of rational functions over the finite field F_q. For a subclass of these functions, the transform yields a polynomial and its factorization as a product of the set of monic irreducible polynomials all of which share a common property P that depends on the choice of rational function. A general formula is derived from the factorization for the number of monic irreducible polynomials of degree n having property P. However it is also possible in some instances to exploit the properties of the factorization to obtain a "closed" form of the answer more directly. We illustrate the method with four examples, two of which appear in the literature. In particular, we give alternative proofs for a result of L. Carlitz on the number of monic irreducible self-reciprocal polynomials and a remarkable result of S. D. Cohen on the number of (r,m)-polynomials, that is, monic irreducible polynomials of the form f(x~r) of degree mr. We also give a generalization of the factorization of x~(q-1) - 1 over F_q that includes the factorization of x~((q-1)~2) - 1. The new results concern translation invariant polynomials, which lead to a consideration of the orders of elements in F_q, the algebraic closure of F_q. We show that there are an infinite number of θ ∈ F_q such that ord(θ) and ord(r(θ)) are related, in the sense that given one, one can infer information about the other.