Number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$

Question

I want to determine the number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$ where $f \in \mathbb{Z}_p[x]$ is an irreducible polynomial with $k$ degree bigger than 2. Is the number of elements always $p^k$? And are the equivalence classes $[f_1],[f_2], \dots, [f_{p^k}]$ always given be the polynomials in $\mathbb{Z}_p$ with degree smaller than $k$?

Answer 1

Yes, you're on the right track of reasoning.

Use the fact that $\mathbb{F}[x]$ has a division algorithm to justify that all possible remainders (=all polynomials of degree strictly less than $k$) form a complete set of distinct residues.

To show that every polynomial is equivalent to a remainder, remember that modding out by $(f)$ is basically saying that $g\equiv h\iff f|(g-h)$. To show that the remainders are distinct, start by writing out what it would mean if two are equivalent, and show it's impossible.

After that, it's easy to count how many remainders! You have $p$ choices for each of the $k$ coefficients of a remainder.