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?

Solution 1:

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 $< d$.

Solution 2:

Hint $ $ Recall $\rm\ R[x]/(f)\:$ has a complete system of reps being the least degree elements in the cosets, i.e. the remainders mod $\rm\:f,\:$ which uniquely exist by the Polynomial Division Algorithm.

Therefore the cardinality of the quotient ring equals the number of such reps, i.e. the number of polynomials $\rm\in R[x]\:$ with degree smaller than that of $\rm\:f.$

Remark $\ $ This is a generalization of the analogous argument for $\rm\:\Bbb Z/m.\:$ The argument generalizes to any ring with a Division (with Remainder) Algorithm, i.e. any Euclidean domain, as explained in the linked answer.