1. QUADRATIC FORMS AND ASSOCIATIVE ALGEBRAS 11
subspace Z just defined for {ei, f \ } . Applying our induction to Z and its subspace
H2T=2 Fei, w e obtain our lemma.
In the setting of the above lemma we call the expression of (1.2a) with 0
m G Z a weak W i t t decompositio n (of V with respect to p); we call it a
W i t t decompositio n if the restriction of p to Z is anisotropic. In particular, if
Z = {0}, we call it a split W i t t decomposition . If (1.2a) is a Witt decomposition,
we call Z a core subspace of V with respect to p, and call dim(Z) the core
dimensio n of (V, p), or simply, of p.
L e m m a 1.4. Suppose p is nondegenerate on V; then the following assertions
hold:
(i) V has a Witt decomposition.
(ii) If (1.2a) is a Witt decomposition and ( is the restriction of ip to Z, then
the integer m and the isomorphism class of (Z, Q are completely determined by
the isomorphism class of'(V, p).
(iii) Every totally isotropic subspace of V is contained in a totally isotropic
subspace of V of dimension m with the integer m determined as in (ii).
P R O O F . There is no problem if p is anisotropic. Suppose V has a nontrivial
totally isotropic subspace X. Clearly X is contained in a maximal totally isotropic
subspace / of V. By Lemma 1.3, we can find a weak Witt decomposition (1.2a) with
/ in place of Y^tLi Fz%- If Z contains a nonzero vector g such that p[g] = 0, then
Fg + I is totally isotropic, which is a contradiction as / is maximal. Therefore p is
anisotropic on Z, so that (1.2a) is a Witt decomposition. This proves (i). Let V =
Z' + ^2^=i(Fe[ + Ffl) be another Witt decomposition; suppose n m. Then, by
Theorem 1.2, we can find an F-linear bijection / of Z onto Z'^Y^lZT(Fei + Ffi)
such that p[xf] = p[x] for every x G Z. Since p is anisotropic on Z, we have
m = n, and at the same time we obtain (ii). Assertion (iii) is also proved, as we
have seen that X C I.
L e m m a 1.5. (i) Given (V, p) with nondegenerate /?, let a and ft be F-linear
injections of a vector space W into V such that p[xa\ = p[xft] for every x G W.
Suppose that p is nondegenerate on Wot. Then there exists an element 7 of O^
such that « 7 = ft. Moreover, such a 7 can be taken from SO^ if dim(F)
dim(VT).
(ii) Given (V, p) as in (i), let h and k be nonzero elements of V such that
p[h] = p[k}. Then there exists an element 7 of O^ such that hj = k. Moreover
such a 7 can be taken from SO^ if p[h] 7^ 0 and dim(V) 1, or if p[h] = 0 and
dim(V) 2.
P R O O F . We can find an F-linear bijection £ of Wa to Wft by (xa)£ = xft
for x G W. Then p[y£] p[y] for every y G Wa. Applying Theorem 1.2 to
V = (Wa) + (Wa)± = (Wft) + (Wft)±1 we can extend £ to an element 7 of 0*\
Then aj = ft. If dim(V) dim(W), we can modify 7 by a suitable element of
O^^WaY) so that det(7) = 1. This proves (i).
As for (ii), suppose p[h] j^ 0; define a, ft : F » V by xa = xh and xft = xk
for x G F. Then our assertion follows from (i). Suppose p[h] 0. Then, by Lemma
1.3, V has weak Witt decompositions V = X + Fh + Fg = Y + Fk + Fi with
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