Let $\alpha$ be a linear operator on a vector space $V$. Let $v_0\in V$ and suppose that the minimal polynomials of $\alpha$ is $m_\alpha(x)$ and of $\alpha$ at $v_0$ is $m_{\alpha,\ v_0}(x)$. Suppose $m_\alpha(x)= m_{\alpha,\ v_0}(x)=[f(x)]^k$, for some monic irreducible $f(x)\in F[x]$.
Using Zorn's Lemma, prove that there exists $W$, an invariant subspace of $V$, such that $V=\left<v_0\right>_\alpha\bigoplus W$
I tried to consider the set $P=\{\,W\mid \text{$W$ is invariant subspace of $V$ and $\left<v_0\right>_\alpha\bigoplus W$}\,\}$ but neither its non-emptiness nor the existence of an upper bound is clear to me. Can someone give me a hint on this? Thank you very much.