Let $f\in \mathbb F_q[x]$ be an irreducible polynomial. Let $e$ be a positive integer. Let $R= \mathbb F_q[x]/(f^e)$. Let $y\in R$.
Prove that if $y^{q-1}-1=0$ then $y\in \mathbb F_q^*$.
Notice that for $e=1$ $R$ is a finite field, so the statement is immediate by the characterization of the subfield $\mathbb F_q$.
Edit: please find the answer below.
Alright, so this is the answer (short version):
by reducing modulo $f$ observe that if $y^{q-1}-1=0$, by standard properties of finite fields, this implies that $y=c+fg$ for some $f,g\in R$.
Now multiply the equation above by $y$ getting $y^q-y$, and plug in the special form for $y$, getting $(fg)^q-fg=fg((fg)^{q-1}-1)=0$ but now $(fg)^{q-1}-1$ is invertible in $R$, so $fg$ must be already zero, which leads to $y=c$.