12 I. QUADRATIC FORMS AND CLIFFORD ALGEBRAS
subspaces X, Y and elements g, L By Theorem 1.2 we can find an element 7 of
O^ such that X 7 = Y, /17 = /c, and gj = £. It X ^ {0}, then we can take 7 from
5 0 ^ by modifying it by a suitable element of 0(p(X).
1.6. Let us now express various things by matrices. Given (V, p) with degener-
ate or nondegenerate /?, take an F-basis {ei}f=1 of V. For x = ^™=1 &e* ^ ^ w ^ n
£i G F, let Xo be the element of F^ whose components are £1, ... , £n. Let /?o
be the n x n-matrix [^(e^, e^)]. . Then tpQ = y?o and ip(x, y) = #0^0 * t?/o- We
call /? o ^
e
matrix that represents p with respect to {e^}^=1, or simply a matrix
representing p when the basis is not specified. Clearly p is nondegenerate if and
only if det(/?o) 7^ 0. Hereafter, we shall often use the same letter for a quadratic
form and the matrix representing it when the basis is fixed. If we change the basis,
then ifo is changed into apQ-la with a G GLn(F). We can define an isomorphism
a ^ a o of GL(V) onto GLn(F) by (xa)o =
XQCXO.
If p is nondegenerate, the map
a ^ a
0
gives an isomorphism of O^ onto the group
(1-3) O(p0) = {Pe GLn(F) I f3p0 •tP = Vo }•
If (W, I/J) = (V, (p) 0 {V', (/?'), and p resp. /?' is represented by po resp. ^Q, then
ij) is represented by diag[(/?0, Po]-
It is an easy exercise to show that every degenerate or nondegenerate p can be
represented by a diagonal matrix. In particular, we have a diagonal representation
of a nondegenerate p with respect to a basis {hi}f=1 if and only if ip(hi, hj) = Q5^
for every i and j with C{ G F x . In such a case we call {hi}^=1 an orthogonal
basis of V with respect to ip.
If V = Yl^iiFfi + ^ e 0 ^s a S P ^ Witt decomposition, we shall often use the
expression (iJm, 2~1r]m) instead of (V, ip), where
' 0 l
m
lm 0
which is twice the matrix representing the quadratic form with respect to the ba-
sis {/1, ... , /
m
, ei, ... , e
m
} . Also we shall often put V =
YHLI
Ffi a n d
^
=
X ^ i ^ei s o t n a t ^m = / ' + /.
Suppose now / ? is nondegenerate; put F
x 2
= { a
2
| a G F
x
} . Take po as above.
Since det(a^
0

tot)
G
det((/?0)Fx2,
the coset
de\J{po)Fx2
in
Fx/Fx2
is completely
determined by (p. We call the coset
det((/?o)Fx2
the discriminant of /?, or of
(V, p), and denote it by d(ip) or d(V, p). If n is odd, then for c e
Fx
we have
d(c(/?) =
cnd(p)
= cd(/). Therefore, given ip with odd n, we can always take
c G
Fx
so that d(cy) is represented by any given element of F
x
. If n is even, by
the discriminant field of p, or of (V, (/?), we understand the extension
(1.5a) K0 =
F({(-l)n'2 det(^o)}1/2)
of F This is either F itself or a quadratic extension of F We then define the
discriminant algebra K of /?, or of (V, ip), by
f if0 if K0 ^ F,
If C is the restriction of £ to a core subspace Z of V, then the discriminant field
and algebra of £ coincide with those of p.
(1-4) »*,
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