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 .
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