I'm just looking for someone to check my work. I am given that I have a matrix with elements in $\mathbb{Q}$ and characteristic polynomial $C(x)=(x^4-1)(x^2-1)=(x-1)^2(x+1)^2(x^2+1)$. I am looking for possible invariant factors, so these must conform to the conditions that for invariant factors $d_i$, $d_i|d_{i+1}$ and $d_1...d_j=C(x)$. So I have that the possibilities for invariant factors are $(x-1), (x-1)(x+1)^2(x^2+1)$ or $(x+1), (x+1)(x-1)^2(x^2+1)$. Is this an exhaustive list? Are these correct?
Thanks.