For the definition of splitting fields of central simple algebras see here. Throughout, is a division algebra with the center and For a subalgebra of we will denote the centralizer of in by

**Theorem**. Let be a subfield of and suppose that Then where

*Proof*. Let As we saw in here, has a structure of a (right) -module and

Note that is a finite dimensional central simple -algebra. In particular, it is primitive. We are now going to show that is a faithful simple -module. First, is faithful because the annihilator of in is an ideal of and, since is a simple algebra, the annihilator is zero. Now, let be a non-zero -submodule of Let and Then, since is an -module, we have Thus is a right ideal of and so Hence is a simple -module. So, by and the structure theorem for primitive rings, we have where To complete the proof of the theorem, we only need to show that Well, we have

and thus

The last equality in is true by the lemma in this post. Therefore

**Corollary 1**. If is a maximal subfield of then is a splitting subfield of In fact, where

*Proof*. If is a maximal subfield, then and

**Corollary 2**. If is a splitting subfield of then is a maximal subfield of

*Proof*. We only need to show that By Theorem 2, where Since is a splitting subfield of we also have where Thus So and Thus and hence because

**Corollary 3**. Let be a subfield of Then the following statements are equivalent.

1) is a maximal subfield of

2) is a splitting subfield of

3) and

4)

*Proof*. Straightforward!