On a theorem of Carlitz

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.