Let T be a linear operator on a finite-dimensional vector space V.Let $\phi(t)$ an irreductible monic divisor of the characteristic polynomial of T, hence for some integer p, $(\phi(t))^p$ is the T-annihilator of a nonzero vector of V.
The part above is a simplified part of what I think is being used as an argument in the proof of the theorem.
I was trying to understand theorem 7.23 from Friedberg's book Linear Algebra. but the highlighted part in the image is not clear to me.I don't understand why a non-zero vector must exist.