Question involving characteristic polynomial of a linear transformation

Question

I was wanting some hints on a question and I have no idea how to approach this:

Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ divides the characteristic polynomial of $T$. Prove that there is a subspace $V'$ of $V$, such that $T(V') \subset V'$ and the restriction $T'=T|_{V'}: V' \rightarrow V'$ has characteristic polynomial equal to $p(x)$.

I know there is a correspondence between the $F[x]$ submodules of $V$ and $T$-stable subspaces of $V$. Where do I go from here? I am really not getting the ramifications of $p(x)$ dividing the characteristic polynomial of $T$.

Thanks!

Answer 1

Hint: Try $V'=\ker p(T)$.

${}{}{}{}{}$