In this post we showed that So the tensor product of two (finite dimensional) division algebras may not even be a reduced algebra let alone a division algebra. There is however the following simple result.

**Theorem**. Let be finite dimensional central division -algebras. If then is a division algebra.

*Proof*. By the corollary in this post, is a finite dimensional central simple -algebra. So

for some finite dimensional central division -algebra and some integer We need to prove that Let We are given that

Now, we have

for some finite dimensional central division -algebras and integers Also, by the theorem in this post,

Thus, using and we have

Hence Similarly and so So divides both and and hence by

See part (2) here.

Advertisements