Let $R_n = F_q[x]/\langle x^n - 1 \rangle$, where $F_q[x]$ is a finite field.
Consider $\mu_a$ which acts on $R_n$ like so; $f(x) \mu_a \equiv f(x^a) \bmod (x^n - 1)$ for $f(x) \in R_n$.
Is this an automorphism?
I'm not too sure where to start here. I know polynomial substitution is a homomorphism, so can I instantly deduce that this is a homomorphism?
If I show it's injective, then I can deduce that it is surjective since $R_n$ is finite, but I'm not sure where to start with injectivity?
Any help would be appreciated!