**Prp1**. Let be a field and The characteristic polynomial and the reduced characteristic polynomial of an element of are equal.

*Proof*. Well, defined by is a -algebra isomorphism. Thus

**Prp2**. (Cayley-Hamilton) Let be a finite dimensional central simple -algebra and Then

*Proof*. Let be a splitting field of with a -algebra isomorphism We have Since is just the characteristic polynomial of we may apply the Cayley-Hamilton theorem from linear algebra to get Thus, since is a -algebra homomorphism, we get

Hence, since is injective, we must have which implies

**Prp3**. Reduced characteristic polynomials are invariant under extension of scalars.

*Proof*. First let’s understand the question! We have a finite dimensional central simple -algebra with We are asked to prove that if is a field extension and then Note that is a central simple -algebra and Now, let be a splitting field of with an -algebra isomorphism But

So we also have an isomorphism Clearly and hence

**Prp4**. Reduced characteristic polynomials are invariant under isomorphism of algebras.

Proof. So are finite dimensional central simple -algebras and is a -algebra isomorphism. We want to prove that for all Let be a splitting field of with a -algebra isomorphism The map is also a -algebra isomorphism. Thus we have a -algebra isomorphism So if then