Cardinality of $R[x]/\langle f\rangle$ via canonical remainder reps. Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$.  How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$?  Secondly, if $R$ is finite with $n$ elements, how do you show that $R[x]/\langle f\rangle$ has exactly $n^d$ cosets?

We will use the following fact (it should have been proved in the above-mentioned class): Let $K$ be any field. Then $K[x]$ is an Euclidean ring, i.e., for every two polynomials $f,g \in K[x]$ such that $g \neq 0$, there exist (unique!) polynomials $q, r$ such that $f = g \cdot q + r$ and $\deg r < \deg g$, with $\deg 0 = -\infty$.
Now, let $\overline{g}$ be any coset of $R[x]/\langle f \rangle$, and $g$ its representative. By division with remainder, we have $g = f \cdot q + r$ with $\deg r < \deg f$ and ... (fill in the gaps here).
For the second part, count the polynomials with degree $< d$.
