Use the division algorithm to prove that in $(\Bbb Z/p\Bbb Z)[x]/\langle f (x)\rangle$ there are at most $p^d$ elements

Question

Suppose that $f(x) ∈ (\Bbb Z/p\Bbb Z)[x]$ is a polynomial of degree $d$. Use the division algorithm to prove that in $(\Bbb Z/p\Bbb Z)[x]/\langle f (x)\rangle$ there are at most $p^d$ elements.

Answer 1

Hint: Any element of $\ \left(\mathbb{Z}/p\mathbb{Z}\right)[x]/\langle f(x)\rangle\ $ has the form $$ g(x)+\langle f(x)\rangle\ , $$ where $\ g(x)\in \left(\mathbb{Z}/p\mathbb{Z}\right)[x]\ $. But if $\ r(x)\ $ is the remainder of $\ g(x)\ $ on division by $\ f(x) $, then (assuming $\ p\ $ is prime) $$ g(x)+\langle f(x)\rangle = r(x)+\langle f(x)\rangle\ , $$ and $\ \deg\left(r(x)\right)< d\ $. Thus, every element of $\ \left(\mathbb{Z}/p\mathbb{Z}\right)[x]/\langle f(x)\rangle\ $ is of the form $$ r(x)+\langle f(x)\rangle $$ for some polynomial $\ r(x)\ $ over $\ \mathbb{Z}/p\mathbb{Z}\ $ of degree less than $\ d\ $. How many such polynomials are there?