I needed the following well-known result in the course I’m teaching:
Lemma. Let be a field and a nilpotent matrix. Then .
The classical proof uses the flag of subspaces
to produce a basis for which is upper triangular. Here is a slick basis-independent commutative algebra proof that shows something better:
Lemma. Let be a commutative ring with nilradical , and let be a nilpotent matrix. Then the characteristic polynomial satisfies
Here we write for the polynomials in of degree smaller than whose coefficients lie in a given ideal .
Note that the formulation should ring a bell: in the previous post we saw that . When is a domain, this reduces to , and the lemma just says that .
Proof of Lemma. We have to show that . Since is nilpotent, there exists with , so . Thus . Evaluating at shows that the constant coefficient is 1.