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.