Page 310 - Advanced Linear Algebra
P. 310
294 Advanced Linear Algebra
Assume now that the theorem is true for dimensions less than and let
dim²= ³~ . Let #= be nonisotropic. Since º #Á #»~º#Á #»£ , Theorem
11.32 implies the existence of a symmetry on for which
=
²#³ ~ #
where . Thus, ~f ~ f on span ²#³ . Since Theorem 11.9 implies that
span²#³ is -invariant, we may apply the induction hypothesis to on
span²#³ to get
O ~
span²#³ Ä $ ~ $
where $ span ²#³ and each $ is a symmetry on span ²#³ . But each $ can
be extended to a symmetry on = by setting # ~ # . Assume that is the
$
extension of to , where ~ on span²#³ . Hence, ~ on span ² # ³ and
=
~ on span²#³.
If ~ , then ~ on = and so ~ , which completes the proof. If
~c , then ~ # on span ²#³ since # is the identity on span²#³ and
=
~ on span²#³. Hence, ~ on and so ~ on .
= #
#
#
The Witt Theorems for Orthogonal Geometries
We are now ready to consider the Witt theorems for orthogonal geometries.
(
Theorem 11.34 Witt's cancellation theorem) Let = and > be isometric
nonsingular orthogonal geometries over a field with char ² - ³ £ . Suppose
-
that
=~ : p : and > ~ ; p ;
Then
: ; ¬ : ;
Proof. First, we prove that it is sufficient to consider the case =~ > . Suppose
that the result holds when =~ > and that ¢ =¦ > is an isometry. Then
²:³ p ²: ³ ~ ²: p : ³ ~ = ~ > ~ ; p ;
Furthermore, : : ; . We can therefore apply the theorem to > to get
: ²: ³ ;
as desired. To prove the theorem when =~ > , assume that
=~ : p : ~ ; p ;
where and are nonsingular and : ; . Let ¢ :¦ ; be an isometry. We
;
:
proceed by induction on dim²:³ .
