On a theorem of Carlitz

TL;DR

This paper simplifies Carlitz's original proof by providing a straightforward construction of a polynomial with key properties, removing the need for a mysterious polynomial and clarifying the underlying reasons for its existence.

Contribution

The paper introduces a simple polynomial construction that replicates Carlitz's key polynomial, clarifying the underlying subtlety and broadening understanding of permutation groups over finite fields.

Findings

A simple polynomial can replace the complex one in Carlitz's proof.

The existence of such polynomials depends on a specific subtlety.

Implications for permutation group generation over finite fields.

Abstract

Carlitz proved that, for any prime power q other than 2, the group of all permutations of the finite field F_q is generated by the permutations induced by degree-one polynomials and x^{q-2}. His proof relies on a remarkable polynomial which appears to have been found by magic. We show here that no magic is required: there is a straightforward way to produce a simple polynomial which has the same remarkable properties as the complicated polynomial in Carlitz's proof. We also identify the crucial subtlety which allows such simple polynomials to exist, and discuss some consequences.