Page 302 - Advanced Linear Algebra

286 Advanced Linear Algebra

fields, the real field and finite fields. Here is a preview of the forthcoming
s
results.
)
-
1 When the base field is algebraically closed, there is an ordered basis 8
for which
v y
x Æ {
x x { {
4~ A Á ~ x {
8
x {
x {
Æ
w z
If = is nonsingular, then 4 8 is an identity matrix and = has an
orthonormal basis.
2 Over the real base field, there is an ordered basis for which
)
8
v y
x Æ {
x {
x {
x {
x x c { {
8
4~ Z Á Á ~ x Æ {
x {
x c {
x {
x {
x {
Æ
w z
3 If is a finite field, there is an ordered basis for which
)
8
-
v y
x Æ {
x {
x {
x {
8
4~ Z Á ² ³ ~ x {
x {
x {
x {
Æ
w z
where is unique up to multiplication by a square and if char ² - ³ ~ , then

we can take ~ .
Now let us turn to the details.
Algebraically Closed Fields

-
If is algebraically closed, then for every  - , the polynomial % c has a
root in , that is, every element of has a square root in . Therefore, we may
-
-
-
(
)
choose ~ °j in 11.2 , which leads to the following result.