A Generalized Counting and Factoring Method for Polynomials over Finite Fields

Abstract

We discuss a transform on the set of rational functions over the finite field ${\mathbb{F}}_q$. 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 ${\mathbb{F}}_q$ 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 ${\overline{\mathbb{F}}}_q$, the algebraic closure of ${\mathbb{F}}_q$. We show that there are an infinite number of $\theta \in {\overline{\mathbb{F}}}_q$ such that $\text{ord}(\theta )$ and $\text{ord}(r(\theta ))$ are related, in the sense that given one, one can infer information about the other.