We will assume that is a field.

**Definition**. Let be a -algebra. A **representation** of is a -algebra homomorphism for some integer If then is called **faithful**.

If is a representation of a -algebra and then is a matrix and so it has a characteristic polynomial. Things become interesting when is a finite dimensional central simple -algebra. In this case, the characteristic polynomial of is always a power of the reduced characteristic polynomial of This fact justifies the name “reduced characteristic polynomial”!

**Theorem**. Let be a finite dimensional central simple -algebra of degree and let Let be a representation of and suppose that is the characteristic polynomial of Then for some integer and

*Proof*. Let be a splitting field of We now define by for all and and then extend it linearly to all We now show that is a -algebra homomorphism. Clearly is additive because is so. Now let and Then

and

So, by Lemma 2, for some integer and

**Example**. Let be a finite dimensional central simple -algebra of degree So and we have a faithful representation defined by for all Let and suppose that is the characteristic polynomial of Then by the above theorem.