I am wondering about the following linear algebra problem:
Let the minimal polynomial of $T$ on a finite-dimensional vector space $V$ be $p^2$, where $p$ is irreducible. Is it true that $V$ contains a proper $T$ invariant subspace?
Is there a relationship between the irreducibility of factors showing up in the minimal polynomial and possible invariant subspaces?
I did find the following: Minimal polynomial is irreducible if the only $T$-invariant subspaces of $V$ are $V$ and $0$
Is there some statement about the converse? Furthermore, I'm not exactly dealing with the converse here, since the minimal polynomial is reducible. I guess, what I can say is, the minimal polynomial splits as a product of irreducible factors (not necessarily linear).