**Introduction**. Let be an algebraic extension. Recall that the **Galois group** of is the set of all automorphisms such that for all The fixed field of is defined by Clearly is a subfield of and If then is called **Galois**. If then we have the following two facts from Galois theory.

**Fact 1**. is Galois is separable and is the splitting field of some polynomial

**Fact 2**. If is separable, then there exists a field extension such that is Galois and

**Notation**. For the rest of this post, is a finite dimensional central division -algebra.

**Note**. By the theorem in this post, has a maximal subfield which is separable over Notice that such maximal subfields need not be Galois over However, as we will see in the following theorem, there exists a finite Galois extension that splits

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

*Proof*. By the theorem in this post, there exists a maximal subfield of such that is separable. Thus, by Fact 2, there exists a field extension such that is Galois and So the only thing left is to prove that is a splitting field of i.e. for some integer By Corollary 1, is a splitting subfield of and so for some integer Thus

I finish this post with a definition. We will look into it in the future.

**Definition**. If has a maximal subfield which is Galois over then we say that is a **crossed product**.