A simple polynomial for a transposition over finite fields

Abstract: Let $q>2$, and let $a$ and $b$ be two elements of the finite field $\mathbb{F}_q$ with $a\ne 0$. Carlitz represented the transposition $(0a)$ by a polynomial of degree $(q-2)^3$. In this note, we represent the transposition $(ab)$ by a polynomial of degree $q-2$. Also, we use this polynomial to construct polynomials that represent permutations of finite local rings with residue field $\mathbb{F}_q$.