List of all $\mathbb{C}[X]$ modules? Let $M$ be a $\mathbb{C}[X]$-Module.
The task ist to list all $\mathbb{C}[X|$-modules $M$ with $\dim_\mathbb{C}M=1$, $\dim_\mathbb{C}M=2$, $\dim_\mathbb{C}M=3$ up to isomorphism.
I'm not sure I understand this question. In general defining a $K[X]$-Module $M$ is equivalent to defining a $K$ linear map $M\longrightarrow M$, so I would just answer "any matrix with 1 component, any $2x2$ matrix, any $3x3$ matrix even with $0$ determinant". This whole thing doesn't make much sense to me, so I'd be glad if anyone could clarify the question.
 A: I will do the case $\dim_{\Bbb C}M=2$ and leave the rest to you.
First note that such a module $M$ is necessarily finitely generated over $\Bbb C[X]$ and hence is of the form $$M=R^n\oplus\bigoplus_{i=1}^kR/(f_i)$$
with $R=\Bbb C[X],0\ne f_i\in R$ monic. Since $\dim_{\Bbb C}R=\infty$ we necessarily have $n=0$. Now we have $$\dim_{\Bbb C}M=\sum_{i=1}^k\deg f_i$$
So by eliminating those $f_i$ that are units, i.e. for which $R/(f_i)=0$, we see that $M$ is of the form $$M=R/(f_1)\oplus R/(f_2)$$ with $\deg f_1=\deg f_2=1$ or $$M=R/(f)$$ with $\deg f=2$. Let us look at the first case: $M$ is of the form $$M=\Bbb C[X]/(X-\alpha)\oplus\Bbb C[X]/(X-\beta)$$
If $\alpha,\beta\in\Bbb C$ are distinct then we can write this as $M=\Bbb C[X]/((X-\alpha)(X-\beta))$ and we are reduced to the second case. If $\alpha=\beta$ we get $$M=(\Bbb C[X]/(X-\alpha))^2$$
So in total we get two types of modules: The first is of the form $(\Bbb C[X]/(X-\alpha))^2$ and the second type is of the form $\Bbb C[X]/((X-\alpha)(X-\beta))$ for $\alpha,\beta\in\Bbb C$. By looking at the annihilators we see that for different choices of $\alpha,\beta$ these modules will not be isomorphic (as $\Bbb C[X]$-modules)
If you prefer to think about these modules as $\Bbb C$-vectorspaces with an endomorphism $f$, the first type correspond to those endomorphisms whose minimal polynomial has degree one (or those that are diagonalizable with exactly one eigenvalue) and the second type correspond to those whose minimal polynomial has degree two (in this case the homomorphism is diagonalizable iff $\alpha\ne\beta$)
