Matrix $A$ satisfies the equality $A^7 = I$

Question

Matrix $A$ satisfies the equality $A^7 = I$. Is it true that the matrix is diagonalizable? Justify the answer.

Answer 1

If we take the field $\Bbb F_7$ and $A = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$, then we see that $A^7 = E$ (the identity matrix) but $A$ is not diagonalisable. (The only eigenvalue is $1$ but $A-I$ has nullity $1$ and not $2$.)

On the other hand, in any field, we have $A = E$ as an example of a matrix that satisfies $A^7 = E$ and is diagonalisable.

Thus, without further information about the field, it seems that one can not conclude anything about the diagonalisability.