Let and suppose that is nilpotent. Show that if commute then
Since is algebraically closed and commute this means that are simultaneously triangularizable, there exists an invertible element such that both and are triangular. Since is both nilpotent and triangular, all its diagonal entries are zero and so the diagonal entries of are the same as the diagonal entries of . Thus,
because , are both triangular and the determinant of a triangular matrix is the product of its diagonal entries. So,
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