Let $R$ be a commutative ring, let $R[X]$ be the polynomial ring over $R$ , and let $a$ be an element of $R$ . Then we can define a homomorphism $\tau_a$ of $R[X]$ by constructing the evaluation homomorphism from $R[X]$ to $R[X]$ taking $r\in R$ to itself and taking $X$ to $X+a$ .

To see that $\tau_a$ is an automorphism, observe that $\tau_{-a}\circ\tau_a$ is the identity on $R\subset R[X]$ and takes $X$ to $X$ , so by the uniqueness of the evaluation homomorphism, $\tau_{-a}\circ\tau_{a}$ is the identity.