Why can a matrix whose kth power is I be diagonalized? Say A is an n by n matrix over the complex numbers so that A raised to the kth power is the identity I. How do we show A can be diagonalized?
Also, if alpha is an element of a field of characteristic p, how do we show that the matrix A=[1, alpha; 0, 1] satisfies A raised to the pth power equals the identity I and cannot be diagonalized if alpha is nonzero.
Please be detailed. I really have no idea how to start on this one.

It can be diagonalized over $\Bbb C$, but not always over $\Bbb R$. Consider a rotation matrix of $2\pi/3$ angles. 
