We show that is not semisimple nor simple, but is a division algebra.

Consider (as a -algebra). Consider as a right -module.

Lemma:

is not semisimple nor simple.

Suppose to the contrary , where are simple -modules (i.e. ). Then there exists nonzero such that has finite order (product of primes). This is impossible in .

Lemma:

as -algebras.

Define where is mapped to , where . Let , .

We can check that is a -algebra homomorphism.

Let . Then , for all . This implies . Hence is injective.

Let . Let , where . . Hence is surjective.

Thus is a division algebra, but is not simple.