Let be a field, , and take such that has degree over . Let be the monic irreducible polynomial over having as a root.
Let be given by . Show that the characteristic polynomial of is .
from the definition of it's easily seen that for any and we have thus if and only if in particular and have the same minimal
polynomials. so is the minimal polynomial of now by, Cayley-Hamilton, the irreducible factors of the characteritsic polynomial and the minimal polynomial of a linear transformation are
the same. thus the caracteristic polynomial of is for some integer comparing the degrees gives us