Diagonalizable linear transformations... Suppose $V$ is a finite dimensional vector space over $\Bbb C$ and that $T: V \rightarrow V$ is a linear transformation such that $T^4=I$. Prove that $T$ is diagonalizable, and that this does not necessarily hold when $V$ is taken over $\Bbb R$. 
I haven't tried much on this since I do not know where to start really. 

*

*The polynomial $x^4-1$ splits into linear factors over $\mathbb C$ and annihilates $T$ so $T$ is diagonalizable over $\mathbb C$.

*The rotation matrix by the angle $\frac{\pi}{2}$ is a counterexample.
