How many conjugacy classes of matrices are there in $\mathop{\mathrm{GL}}_{10}(\mathbb{Q})$ with minimal polynomial $X^7 − 4X^3$?
I initially thought that since $0$ is a root of the minimal polynomial, it is also an eigenvalue. But if a matrix has $0$ as an eigenvalue, it has a nontrivial kernel, and hence is not invertible. In particular, it is not in $\mathop{\mathrm{GL}}_{10}(\mathbb{Q})$, so the answer to the question above is that there are no such conjugacy glasses?
As this was posed as a supposedly hard exam question, I find it hard to believe that the solution should be this simple - where does the argument above fail?