Let $F$ be a field and $V$ a finite-dimensional vector-space over $F$. Let $T : V \longrightarrow V$ be a linear transformation, such that for every $v \in V$, there exists $n \in \mathbb{N}$ such that $T^n(v) = v$.\\
(A) Show that if $F = \mathbb{C}$, then $T$ is diagonalizable.\\
(B) Show that if $\operatorname{char}(F) > 0$, then there exists a non-diagonalizable $T$ satisfying the above hypothesis.