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 .

This suggests that we shouldn’t work with but with its anadrome (or reciprocal) .

*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.