The goal is to prove that if is a division ring which is finite dimensional, as a vector space, over its center , then there exist two elements such that Here means the -algebra generated by We begin with a standard field theory result.

**Lemma**. (Artin) Let be a field extension with Then is simple if and only if the number of intermediate fields is finite.

From now on, is a noncommutative division algebra with the center and

**Theorem**. (Herstein and Ramer*, *1972) Let be a sub-division algebra of and a maximal subfield of which is a simple extension of There exists such that

*Proof*. By the lemma there exist only a finite number of intermediate subfields which we’ll call them Let be the centralizer of in . Clearly for all Note that is an infinite field because otherwise would be a finite division algebra and thus, by the Wedderburn’s little theorem, would have to be a field, which is false because we assumed that is noncommutative. Therefore cannot be equal to the union of finitely many of proper vector subspaces. Hence there exists some such that

Now suppose that the claim in the theorem is false. Then for all and hence is one of the Since is infinite, the set is infinite. But the number of is finite and thus there exist distinct elements such that for all Choose Then there exist such that for all Playing with these relations, with the fact that will eventually give us Thus is in the centralizer of But for some because Thus which is not possible by

Recall that has a maximal subfield which is a simple extension of (see the theorem in this post). If , then obviously is a maximal subfield too. .

**Corollary**. There exist such that where

*Proof.* Choose such that is a maximal subfield of By the theorem, there exists such that Let where Since every subalgebra of is algebraic over and hence it is a division ring. So is a division ring. Let be the center of . Clearly Now let Then, since and are both maximal subfields of and every maximal subfield of a division algebra contains the center, Thus So, since is a maximal subfield of we have But is also a maximal subfield of and hence Therefore and so

The above corollary is known as “Albert’s theorem”. For another proof, see Corollary 15.17 in Lam’s book “A first course in noncommutaive ring theory”.