If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less than or equal to $n$ and $x^nP$ to be the set of these polynomials that are multiples of $x^n$, then are the quotient spaces $\mathbb{F}[x]/N$, $\mathbb{F}[x]/E$ and $\mathbb{F}[x]/x^nP$ finite dimensional?
I do see how to get a basis for a quotient space from one for the space and one for its subspace but I can't decide whether or not they're finite dimensional, especially for these examples! Could you help me by treating one of these cases in details to I can see where I misunderstand?
Thank you very much!
We have that
$$\dim_{\Bbb F}\Bbb F[x]=\infty\;,\;\;\dim_{\Bbb F}N=n+1\implies \dim_{\Bbb F}\Bbb F[x]/N=\infty$$
For $\;\Bbb F[x]/E\;$ , we have that
$$x+E\;,\;\;(x+x^3)+E\;,\ldots,(x+x^{2n-1})+E\;,\ldots$$
are all linearly independent in the quotient space, since
$$\sum_{k=0}^na_k\left((x+x^{2k-1})+E\right)=\overline 0 (=E)\;,\;\;a_k\in\Bbb F\iff$$
$$\sum_{k=0}^n(a_kx+a_kx^{2k-1})=\left(\sum_{k=0}^na_k\right)x+\left(\sum_{k=0}^na_kx^{2k-1}\right)\in E\iff$$
$$\sum_{k=0}^na_k=0\;,\;and\;\;a_k=0\;\;\forall\,k$$
Now you try some attack on the third case.