Page 305 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 289

Z
Z
Z
Z
span²" ÁÃÁ" ³ q span²# ÁÃÁ# Á' ÁÃÁ' ³ ~ ¸ ¹

Z
Z

and so
Z
b ² c ³ 
Z
that is,  Z . By symmetry,  and so ~ Z . Finally, since ~ Z , it
follows that ~ Z .
Finite Fields
To deal with the case of finite fields, we must know something about the
distribution of squares in finite fields, as well as the possible values of the
scalars º#Á #» .
- be a finite field with elements.

Theorem 11.27 Let
)
1 If char²- ³ ~ , then every element of - is a square.

)
2 If char²- ³ £ , then exactly half of the nonzero elements of - are

squares, that is, there are ² c ³° nonzero squares in - . Moreover, if %
is any nonsquare in - , then all nonsquares have the form % , for some
 - .
, let - i be the subgroup of all nonzero elements in and
-
Proof. Write -~ -
let
i
i

²- ³ ~ ¸ “  - ¹
be the subgroup of all nonzero squares in - . The Frobenius map

i i ¢- ¦ ²- ³ defined by ² ³ ~ is a surjective group homomorphism, with
kernel

ker² ³ ~ ¸  - “ ~ ¹ ~ ¸c Á ¹
i
(

If char²-³~ , then ker ² ³~¸ ¹ and so is bijective and - ( ( i ~ ²- ³ , (

i
(ker
)
which proves part 1 . If char²-³ £ , then ( ² ³ ~ and so - ( ( i ~ ²- ³ , (

(
)
which proves the first part of part 2 . We leave proof of the last statement to the
reader.
Definition A bilinear form on is if for any nonzero  -universal there
=
exists a vector #  = for which º#Á #» ~ .
Theorem 11.28 Let be an orthogonal geometry over a finite field with
=
-
char²-³ £ and assume that has a nonsingular subspace of dimension at
=
least . Then the bilinear form of is universal.
=

Proof. Theorem 11.21 implies that contains two linearly independent vectors
=
" # and for which
º"Á"» ~ £ Á º#Á#» ~ £ Á º"Á#» ~