Given $C$ a field, $f$ a polynomial in $C[x]$ and $I = (f)$, describe $C[x]/I$ In my course it has been proven that the set $I_f$ whose elements are of the form $r+I$ ,where $r\in C[x]$ such that $r=0$ or $\deg(r) < \deg(f)$, is in bijection with $C[x]/I$.
I wonder whether it is possible to state a stronger fact: $I_f = C[x]/I$.
It may follow from the fact that $C[x]/I$ is an Euclidean Domain; if $g\in C[x]/I$ then there exist $h,r\in C[x]/I$ such that $g=fh+r$ and $r=0$ or $\deg(r)<\deg(f)$, implying that $g+I=fh+r+I=r+I$.
Is this observation sufficient to state that any class in $C[x]/I$ has at least one representative and thus to prove that $C[x]/I = I_f$?
In addition, if the field is finite and $n=|C|$ is it true that $|I_f|=n^{\deg(f)}$ ? 

Let $f(x)\in \Bbb C[x]$ and $I=(f(x))$. 

Consider $\psi: \Bbb C[x]\to \frac{\displaystyle \Bbb C[x]}{\displaystyle I}$, defined by $\psi(g(x))=g(x)+I$. 

Here $\psi$ is a surjection.
