Modules over $\mathbb Q[x]$ that are $2$-dimensional as vector spaces over $\mathbb Q$

Question

I would like to find all $\mathbb Q[x]$-modules that are $2$-dimensional as vector spaces over $\mathbb Q$.

I do not even know where to start. Answers or any suggestions would help. Thanks.

Answer 1

If $V$ is a two-dimensional $\mathbb Q$-vector space, then a $\mathbb Q[x]$-module structure on $V$ is given precisely by specifying how $x$ operates on $V$. The only condition is that $v\mapsto xv$ is linear. So what you are looking for are twodimensional spaces together with an endomorphism.

Answer 2

In general, a module over $\mathbb{Q}[x]$ is nothing but a $\mathbb{Q}$-vector space $V$ equipped with a linear map $T:V\to V$ ($T$ should be thought of as the action of $x$ on $V$). The same statement is true if $\mathbb{Q}$ is replaced by any field.

For example, here's something that might be natural to you: the complex numbers $\mathbb{C}$ is a two-dimensional vector space over $\mathbb{R}$. We can also think of $\mathbb{C}$ as an $\mathbb{R}[x]$-module by letting $x$ act on $\mathbb{C}$ as the imaginary unit $i$ would by multiplication. In other words, $\mathbb{C}$ is the $\mathbb{R}[x]$-module obtained by specifying the $\mathbb{R}$-linear map $T:\mathbb{C}\to \mathbb{C}$ given by rotation by $90$ degrees. Alternatively, we know that $\mathbb{C}\cong \mathbb{R}[x]/(x^2+1)$ as $R$-algebras (via the map $a+bx\to a+bi$ for $a,b\in\mathbb{R}$) and the $\mathbb{R}[x]$-module structure on $\mathbb{C}$ is equivalent to the $\mathbb{R}[x]$-module structure on $\mathbb{R}[x]/(x^2+1)$ under this isomorphism.

Exercise 1: If $V$ is a $\mathbb{Q}[x]$-module, then there is a ring homomorphism $\mathbb{Q}[x]\to \text{End}(V)$ (where $\text{End}(V)$ is the ring of endomorphisms of $V$ as a $\mathbb{Q}$-vector space; it's also isomorphic to the ring of $n\times n$-matrices with coefficients in $\mathbb{Q}$ if $V$ is finite dimensional with dimension equal to $n$). Prove that the kernel of this ring homomorphism is the ideal generated by the minimal polynomial of the linear map $V\to V$ determined by multiplication by $x$.

Challenging Exercise: If you'd like a challenge, then here's one. Prove the standard theorem on the Jordan canonical form using the structure theorem for finitely generated modules over a PID (in this case, the PID would be $\mathbb{C}[x]$). (Hint: If we're studying a linear transformation of a finite dimensional $\mathbb{C}$-vector space, then we're really just studying a finitely generated module over the PID $\mathbb{C}[x]$.)

I hope this helps!