**Introduction**. We first need to recall a few facts from basic field theory. For the definition of separability see the introduction section in this post. We have the following facts.

**Fact 1**. Let be an algebraic field extension and Then is separable if and only if is separable over

**Fact 2**. Let be a chain of fields and suppose that is algebraic. If both and are separable, then is separable.

**Fact 3**. If is a separable field extension and then for some

**Theorem**. Let be a division algebra with the center and suppose that There exists such that is a maximal subfield of and is separable.

*Proof*. Let be the set of all subfields of which are separable extensions of This set is non-empty because Since the set with has a maximal element, say Let be the centralizer of in Suppose that Then is a noncommutative division ring and Let be the center of Then, since is commutative, we have by the double centralizer theorem. Clearly is algebraic over because Hence, by the Jacobson-Noether theorem, there exists such that is separable over So we have the chain of fields where both and are separable (see Fact 1). Thus, by Fact 2, is separable and so But this contradicts the maximality of in This contradiction implies and so, by Corollary 3, is a maximal subfield of Finally, by Fact 3, for some