# How would you find the characteristic polynomial of a nilpotent matrix with complex coefficients?

I'm trying to find the characteristic polynomial of an nxn nilpotent matrix with complex coefficients. I understand how to do it with standard coefficients but I'm a bit confused as to how to proceed with complex coefficients.

it is $$x^n$$ while the minimal polynomial is some $$x^k$$ with $$1 \leq k \leq n,$$ unless your matrix is the zero matrix