Page 482 - Advanced Linear Algebra

466 Advanced Linear Algebra

Theorem 18.11 (Wedderburn's theorem ) If is a finite division algebra, then
+
+ is a field.
Proof. We must show that is commutative. Let . ~ + i be the multiplicative
+
group of all nonzero elements of . The class equation is
+
i
i

( i (+~ A²+ ³ b ²+ ¢ *² ³³
(
(
where the sum is taken over one representative from each conjugacy class of

size greater than . If we assume for the purposes of contradiction that is not
+

i
commutative, that is, that A²+ ³ £ + i , then the sum on the far right is not an
empty sum and so ( i (*² ³ ( i ( + for some  + i .

The sets A²+³ and *² ³ are subalgebras of + and, in fact, A²+³ is a
commutative division algebra; that is, a field. Let ( (A²+³ ~' ‚ . Since
A²+³ ‹ *² ³, we may view *² ³ and + as vector spaces over A²+³ and so

( (*² ³ ~ ' ² ³ ( and + ~ '
(

for integers  ² ³ . The class equation now gives

'c

'c ~ ' c b
' ² ³ c

² ³
and since ' ² ³ c “ ' c , it follows that ² ³ “ .

If 8²%³ is the th cyclotomic polynomial, then 8²'³ divides ' c . But

8²'³ also divides each summand on the far right above, since its roots are not

roots of ' ² ³ c . It follows that 8 ² ' ³ “ ' c . On the other hand,
8²'³ ~ ²' c ³


+
and since  + implies that ' ( c ( € ' c , we have a contradiction. Hence
i
A²+ ³ ~ + and + is commutative, that is, + is a field.
i
Finite-Dimensional Real Division Algebras
We now consider the finite-dimensional division algebras over the real field .
s
In 1877, Frobenius proved that there are only three such division algebras.
Theorem 18.12 (Frobenius, 1877 ) If is a finite-dimensional division algbera
+
over , then
s
+~ Á +~ d +~ i or
s
Proof. Note first that the minimal polynomial ²%³ of any  + is either

linear, in which case  s or irreducible quadratic ²%³ ~ % b % b with

c  . Completing the square gives