Throughout, is a field and is a finite dimensional central simple -algebra of degree

If is a splitting field of then, by definition of splitting fields, there exists a -algebra isomorphism Now let and put i.e. the characteristic polynomial of in The goal is to prove that and does not depend on or We will then call the reduced characteristic polynomial of

**Notation**. Let and be field extensions and suppose that is a -algebra homomorphism. Note that, since is a field, is injective. We now define the map by Clearly is a -algebra injective homomorphism.

**Lemma 1**. Let and be splitting fields of with a -algebra homomorphism Suppose that is an -algebra isomorphism. There exists an -algebra isomorphism such that

*Proof*. We will apply the notation and Theorem 2 in this post. By that theorem there exists an -algebra isomorphism such that for all Thus

**Lemma 2.** Let be a splitting field of and Let be a -algebra isomorphism and let be any -algebra homomorphism. Then for some integer and for all

*Proof*. Since is simple, is injective and so is a central simple -subalgebra of Thus, by the theorem in this post, there exists a central simple -algebra such that

Let Then, since we’ll get from the above isomorphism that and so Now, let’s define a map by

for all Note that is repeated times in because Clearly is an -algebra homomorphism because is so. Thus, by the Skolem-Noether theorem (see Corollary 2), there exists an invertible matrix such that for all Thus if then and are similar and so their characteristic polynomials are equal. It now follows from that

**Corollary**. Let be a splitting field of and If are -algebra isomorphisms, then

To be continued in part (2).