Is a non-zero linear map an automorphism if its characteristic polynomial is irreducible?

Question

Let $V$ be a finite-dimensional vector space over a field $F$ and let $T: V \rightarrow V$ be a linear map, that isn't the zero function. Assume the characteristic polynomial $p_{T}(X)$ of $T$ is irreducible over $F$. Consequently, a proper, non-trivial, T-invariant subspace $U$ of $V$ cannot exist. I.e. the kernel of $T$ must be the trivial subspace, thus $T$ is injective. In particular, since $T \in End(V)$, it follows that $T$ is a vector space automorphism. In short: $(T\neq0 \ \land p_{T}(X) \ is \ irreducible ) \implies T \in Aut(V). \ $ Is this correct? Also, wouldn't this imply that every non-zero element $v \in V$ is a cyclic vector for $T$?