Let be a field, an extension of of finite degree, and let . Show that if is the matrix of the linear transformation corresponding to ‘multiplication by ‘ (described here) then is a root of the characteristic polynomial of . Use this result to obtain monic polynomials of degree 3 satisfied by and .
Let be the -linear transformation described here. If is the characteristic polynomial of , then we have . On the other hand, , and so since is injective. So is a root of .
Consider the basis of over . Evidently, with respect to this basis, the matrix of is . As we showed previously, the characteristic polynomial of is . So satisfies . (Surprise!)
Similarly, has the matrix . Evidently, the characteristic polynomial of is . We can verify that actually satisfies this polynomial (WolframAlpha agrees.)