## In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.

## Problem 91

Show that the matrix $A=\begin{bmatrix}

1 & \alpha\\

0& 1

\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$.

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