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I'm working on a homework question, and I'm stuck. The question is:

Let $A$ and $B$ be $2n \times 2n$ rational matrices with $A^2=B^2=-I$.

The first part of the question asks to show that $A$ and $B$ are similar, and that the transition matrix is rational. I believe I've done that. However it's this second part that has me stumped:

Suppose $A$ and $B$ have integer coefficients. Can we assume that $C$ and $C^{-1}$ ($C$ is the transition matrix) have integer coefficients as well?

The hint given was to convert $\mathbb{Z}^{2n}$ into a $\mathbb{Z}[x]$-module.

While I'm sure there are other methods to doing this, I'm interested in following the direction of the hint, it seems like an interesting method, and I would like to get better at using module theory as a practical tool.

What I've Done So Far: So for starters, in order to make $\mathbb{Z}^{2n}$ into a $\mathbb{Z}[x]$-module, we need a linear map. But it seems like there is a very natural choice in this case, namely, the map $T$ underlying the similar matrices $A$ and $B$. However, I'm not sure where to go from here. The only thing I can think of it using the structure theorem for finitely -generated modules over a PID, but $\mathbb{Z}[x]$ is not a PID, so that won't work.

1 Answers 1

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First of all, note that you have two ways of making $\mathbb{Z}^{2n}$ into a $\mathbb{Z}[x]$-module: one using the action of $A$, and one using the action of $B$. The matrices $A$ and $B$ are similar over $\mathbb{Z}$ if and only if the two modules are isomorphic (exercise). The way you have written it, it seems like you are already assuming that the two definitions will give you the same module.

Now, you want to apply the theory of modules over PIDs, but $\mathbb{Z}[x]$ is not a PID. However note, that you are given the extra information that $A^2=B^2 = -I$. So when you define your $\mathbb{Z}[x]$-module by specifying that $x$ acts through $A$, say, then you know that this action factors through $\mathbb{Z}[x]/(x^2+1)$, since $x^2+1$ acts trivially. So both with the $A$-action and the $B$-action, you are actually defining $\mathbb{Z}[x]/(x^2+1)$-modules. But what is this quotient?

I will let you take it from there.

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    As far as I can tell, my problem seems to lie in an incomplete understanding of the structure theorem, namely translating back and forth between statements about modules and statements about matrices.2012-04-09