These two questions are the exercises in Hungerford's Algebra(p.462 ex.1 and ex.5).
1. If A is afinite dimensional central simple algebra over the field K, then
A tensor the opposit ring of A over K is isomorphic to a space of n by n
matrix over K. Where n is the dimension of A over K.
2. If A is a finite dimensional central simple algebra over a field K, then
the dimension of A over K is a perfect square.
Thank you!!!
define the mapOriginally Posted by chipai
by
for all
this map is K-bilinear and so it induces a K-module homomorphism
defined by
for all
(c \otimes d))(x)=f(ac \otimes db)(x)=acxdb=f(a \otimes b)(cxd)=f(a \otimes b)f(c \otimes d)(x).)
alsois injective because
is a (central) simple algebra and so
which is an ideal of
must be trivial. finally
=\dim_K \text{End}_K (A))
and therefore
Note: some facts that are used here:
1)for any
dimensional
vector space
we know this from elementary linear algebra.
2) the tensor product of a central simple algebra and a simple algebra is a simple algebra. also the tensor product of two central simple algebras is central simple.
3) for any finite dimensionalwe have
(\dim_K W).)
finally note thatcannot be replaced by
because then
first note that if2. If A is a finite dimensional central simple algebra over a field K, then
the dimension of A over K is a perfect square.
Thank you!!!is any field extension, then
is an
-algebra and
now if we assume that
then
is a finite dimensional simple algebra. thus
for some finite dimensional
and some positive integer
sincewe must have
thus
and therefore
=n^2.)
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