Throughout is a field, is the algebraic closure of and is a finite dimensional central simple -algebra.

**Lemma. ** for some integer

*Proof*. Let By the first part of the corollary in this post we know that is simple. We also have

It is easy to see that if is a -basis for then is an -basis for Thus So is a finite dimensional central simple -algebra and hence, since is algebraically closed, for some by Remark 2 in this post.

**Theorem**. If is a finite dimensional central simple -algebra, then is a perfect square.

*Proof*. By the lemma, there exists an integer such that Thus

**Definition**. The **degree** of is defined by

**Remark**. Let be a finite dimensional -algebra. Then is reduced if and only if is a finite direct product of finite dimensional division -algebras. In this case, for some integer

*Proof*. obviously is (left) Artinian because and so is nilpotent. Thus because is reduced and so is semisimple. The result now follows from the Artin-Wedderburn theorem. The converse is trivial. Finally, the fact that, by the above theorem, is a perfect square for any finite dimensional division algebra proves the last part of the remark.