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 t pQ = 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 F fi 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 t ot) 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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