1. QUADRATIC FORMS AND ASSOCIATIVE ALGEBRAS 13 L e m m a 1.7. Suppose that dim(V) = 2 and p is nondegenerate. Then Lp is isotropic if and only if d((p) is represented by —1. P R O O F . If tp is isotropic, then by Lemma 1.3, V has a split Witt decomposition, and so d(y) is represented by — 1. To prove the converse, take an orthogonal basis {x, y} of V put a = ip[x] and b = (f[y]. Suppose —1 G d(^) then ab = —c2 with c G F x , and (/?[fcc -f- cy] = 0, so that if is isotropic. This proves our lemma. 1.8. By an algebra over a field F, or simply by an F-algebra, we understand an associative ring A which is also a vector space over F of finite dimension such that (ax)(by) = abxy for a, b G F and x, y € A. If A has the identity element I At then identifying a with QAA f° r every a G A, we can view F as a subring of A. In such a case the identity element of F can be identified with I A , and we denote it simply by 1. Such an A is called a division algebra if every nonzero element of A is invertible. A subring of A that is a vector subspace of A over F is called a subalgebra. In this book we always assume that an F-algebra is of finite dimension over F. We call an F-algebra A with identity element simple if it has no two-sided ideals other than {0} and A itself. Wedderburn's theorem says that A is simple if and only if it is isomorphic to a matrix algebra Mrn(D) with a positive integer m and a division algebra D over F. Moreover, the isomorphism class of D is determined by A. We call A central over F if F is the center of A. If A = Mm(D), then the center of A coincides with the center of D, which is clearly a field. Thus a simple algebra is central simple over its center. Let K be an arbitrary extension field of F. Then A is central simple over F if and only if A ® F K is central simple over K. An algebra is called semisimple if it is the direct sum of finitely many simple algebras. Let F be the algebraic closure of F. Then A is central simple over F if and only if A Sp F is isomorphic to a matrix algebra Mn(F) with some positive integer n. If such an A is isomorphic to Mm(D), then m divides n, [D : F] = (n/ra) 2 , and [A : F] = n2. If A ®j? F is semisimple, then A is semisimple, but the converse is not necessarily true. Given an F-algebra A with identity element, by an involution of A we under- stand a bijection p of A onto itself such that (x -f y)p = xp + yp, (xy)p — ypxp, and (xp)p = x, where x p denotes the image of x under p. We do not necessarily assume that p is F-linear. The restriction of p to F is an automorphism of F of order 1 or 2. Given an involution p of A, we can define an involution x i— £ x p of Afn(A) by (***%• - (x3l)p. 1.9. Given a central simple algebra A over F, define ra G End(A, F ) for a G ^4 by XTa — xa for x G A. Then a i— » r a is an F-linear ring-injection of ^4 into End(^4, F ) . Taking an indeterminate t, denote by fa(t) the characteristic polynomial of ra. Fix an isomorphism a : ^4 (g)^ F — Mn(F). Then we easily see that r a as an element of End(A ®F F , F ) is represented by diag[cr(a), .. . , cr(a)] with j (a) repeated n times. Let ?a(£) be the characteristic polynomial of a (a). Then fa(t) = ga(t)n Now one can prove that ga has coefficients in F. (This is easy if the characteristic of F does not divide n, since fa has coefficients in F, but the fact is true regardless of the characteristic of F ) Clearly ga(a) = 0. We call ga the

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