Let be an matrix over . Show that if for some , then is diagonalizable.
Let be a field of characteristic . Show that has finite order but cannot be diagonalized over unless .
Since , the minimal polynomial of over divides . In particular, the roots of are distinct. Since contains all the roots of unity, by Corollary 25 on page 494 of D&F, is diagonalizable over .
Note that . By an easy inductive argument, then, , and in particular, .
Suppose . Now is in Jordan canonical form, and is not diagonalizable. (See Corollary 24 on page 493 of D&F.) So cannot be diagonalizable, for if it were, then so would . (If is diagonal, then so is .)