Is $\mathbb{Z}_p [x]$ finite? Seems to me like it is.  There are only finitely many distinct powers of $x$ modulo $p$, by Fermat's Little Theorem (they are $\{1, x, x^2, ...,  x^{p-2}\}$), and the coefficient that I choose for each of these powers can only be taken from $\{0,1,2,..., p-1\}$.  So essentially I'm choosing amongst $p$ things $p-1$ many times, resulting in at most $p^{p-1}$ distinct polynomials. 
Yet an assignment claims that $\mathbb{Z}_p[x]$ is infinite. 

If $p=3$, it's not true that $x=x^4$ in this ring $\mathbb Z_3[x]$. Only the coefficients are mod $p$. While $x^{p-1}$ might be the same as $1$ as a function, at least on $\mathbb Z_p^*$, it's not the same as a symbol which represents multiplying an indeterminate $x$ by itself $p-1$ times.
