Page 306 - Advanced Linear Algebra
P. 306
290 Advanced Linear Algebra
Given any - , we want to find and for which
~ º " b#Á " b#» ~ b
or
~ c
((
If ( ~ ¸ -¹ , then ( ~ ² b ³° , since there are ² c ³° nonzero
squares , along with ~ . If ) ~¸ c -¹ , then for the same
reasons (( ~ ² b ³° . It follows that ( q ) cannot be the empty set and so
)
there exist and for which ~ c .
Now we can proceed with the business at hand.
-
Theorem 11.29 Let = be an orthogonal geometry over a finite field and
assume that is not symplectic if char ² - ³ ~ . If char ² - ³ £ , then let be a
=
-
fixed nonsquare in . For any nonzero - , write
v y
x Æ {
x {
x {
x {
?² ³ ~ x {
x {
x {
x {
Æ
w z
where rk²? ² ³³ ~ .
)
8
1 If char²-³ ~ , then there is an ordered basis for which 4 ~ ? ² ³ .
8
)
8
2 If char²-³ £ , then there is an ordered basis for which 4 8 equals
?² ³ or ?² ³.
)
(
Proof. We can dispose of the case char²-³ ~ quite easily: Referring to 11.2 ,
-
since every element of has a square root, we may take ~ ² j ³ c .
If char²-³ £ , then Theorem 11.21 implies that there is an ordered orthogonal
basis
8 ~ ²" Á ÃÁ" Á' ÁÃÁ' ³
for which º" Á " » ~ £ and º'Á '» ~ . Hence, 4 8 has the diagonal form
v y
x Æ {
x {
x {
4~ x 8 {
x {
x {
Æ
w z
