Let $V$ be a finite dimensional vector space and $T$ be some linear operator.
Suppose the minimal polynomial has a factor $(x-c)^2$. If $T$ has Jordan form, then we can assert that the biggest jordan block corresponds to $c$ has size $2 \times 2$.
Now, my question is, in arbitrary field one does not always have Jordan decomposition, so if we look at its rational form, what does this power mean? More precisely, suppose the minimal polynomial of $T$ is of the form $q^np$ where $q$ is irreducible and $q$ and $p$ are relatively prime, can one conclude anything from the power?
I hope my question is not too vague. Once we are not working on $\mathbb{C}$, I get somewhat confused about rational form.