Page 478 - Advanced Linear Algebra

462 Advanced Linear Algebra

² ³ c ~ ² c ³ IJ c ³

and since the left-hand side is not invertible, neither is one of the factors
c , whence  Spec ² ³. But

² ³ c ~

and so ~ ² ³  ²Spec ² ³³ . Hence, Spec ² ² ³³ ‹ ²Spec ² ³³ .

Theorem 18.9 Let be an algebra over an algebraically closed field . If
(
-
Á  (, then
Spec² ³ ~ Spec² ³
Proof. If £ ¤ Spec ² ³ , then c is invertible and a simple computation
gives
c
² c ³´ ² c ³ c µ ~
and so c is invertible and ¤ Spec ² ³ . If ¤ Spec ² ³ , then is
invertible. We leave it as an exercise to show that this implies that is also
invertible and so ¤ Spec ² ³ . Thus, Spec ² ³ ‹ Spec ² ³ and by symmetry,
equality must hold.

Division Algebras
Some important associative algebras ( have the property that all nonzero
elements are invertible and yet is not a field since it is not commutative.
(

-
Definition An associative algebra + over a field is a division algebra if
every nonzero element has a multiplicative inverse.
Our goal in this section is to classify all finite-dimensional division algebras
over the real field , over any algebraically closed field and over any finite
s
-
field. The classification of finite-dimensional division algebras over the rational
field is quite complicated and we will not treat it here.
r
The Quaternions
Perhaps the most famous noncommutative division algebra is the following.
Define a real vector space with basis
i
8 ~ ¸ Á Á Á ¹
To make into an -algebra, define the product of basis vectors as follows:
-
i
)
1 % ~ % ~ % for all %  8

2) ~ ~ ~ c
3) ~ Á ~ Á ~
4) ~c Á ~c Á ~c