Throughout this post, is a field and is a finite dimensional central simple -algebra.

**Definition**. A field is called a **splitting field** of if and as -algebras, for some integer

**Remark**. By this lemma, the algebraic closure of is a splitting field of any finite dimensional central simple -algebra. Also, if f is a splitting field of then and so

**Theorem 1**. Let where is some finite dimensional central division -algebra and let be a field containing Then is a splitting field of if and only if is a splitting field of

*Proof*. Let Then and so If is any field containg then

Now, suppose that is a splitting field of Then Also, since is a finite dimensional central simple -algebra, for some division ring and some integer Therefore, by we have and thus and Hence Conversely, if is a splitting field of then and gives us

**Corollary**. There exists a finite Galois extension such that is a splitting field of

*Proof*. Trivial by Theorem 1 and the theorem in this post.

**Notation**. Let and be field extensions and suppose that is a -algebra homomorphism. Note that, since is a field, is injective. Now, given an integer we define the map by Clearly is a -algebra injective homomorphisms.

**Theorem 2**. Let and be field extensions and suppose that is a -algebra homomorphism. Suppose also that is a spiltting field of with an -algebra isomorphism Then is a splitting field of and there exists an -algebra isomorphism such that for all

*Proof*. The map is an isomorphism and so

Therefore

So we have an isomorphism It is easy to see that for all