Generalization of a Theorem of Carlitz

TL;DR

This paper extends Carlitz's theorem by deriving explicit formulas for counting irreducible polynomials obtained through quadratic transformations over finite fields, using combinatorial methods and Hurwitz genus formula.

Contribution

It introduces a generalization of Carlitz's result, providing explicit formulas for a broader class of irreducible polynomials over finite fields.

Findings

Explicit formulas for irreducible polynomials via quadratic transformations

Application of combinatorial arguments and Hurwitz genus formula

Extension of Carlitz's original theorem

Abstract

We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic transformation. Our main tools are a combinatorial argument and Hurwitz genus formula.