14 I. QUADRATIC FORMS AND CLIFFORD ALGEBRAS reduced polynomia l of a. We also define a multiplicative map NA/F : A and an F-linear map T r ^ / F : A — F by F (1.6) A ^ / F ( a ) = det [(7(a)], T r A / F ( a ) = tr[r(c ( a G A ) . We call these the reduce d nor m and the reduced trace of a. Clearly NA/F(c) = c n and TYA/F(C) — nc for c G F. Let A be a semisimple F-algebra given as A = @ [ = 1 -A* with simple F-algebras v4^. Let Ki be the center of A\. Then F^ is a finite algebraic extension of F, so that NKxiF and Tr^ z / / F are meaningful. Now we define NA/F, T r ^ / F : A — » F by r r (1.7) iV A / F (a) = n ^ / ^ (NAi/*i ( a *))' T r A/F(a) = ^ T r K i / F (TvAi/Ki (a,)) 2 = 1 2 = 1 for a = (ai, .. . , ar) G A with di £ Ai. These are also called the reduced nor m and the reduced trace of a. Now it can be shown that the F-bilinear form (1.8) (x, y) H- TrA/F(xy) is nondegenerate, if and only if Ki is separable over F for every i. Since no insepa- rable algebraic extension will appear in this book, a bilinear form of the above type will always be nondegenerate. In particular, if A = M2(F), then clearly (1.9) NA/F(°) — det (a) and TrA/F(a) = tr(a) 1.10. There is one special type of algebra called a quaternion algebra over a field F, which appears often in our treatment. By this we understand a central simple algebra A over F such that [A : F] = 4. By Wedderburn's theorem, a quaternion algebra over F is either a division algebra or isomorphic to M2(F). In this book we assume that F has characteristic ^ 2 whenever we speak of a quaternion algebra over F For such an A we have clearly (1.10) ga(t) =t2- TrA/F(a)t + NA/F(a) (a G A). Let us now show that there exists an F-linear involution L of A such that ( i . i i ) NA/F(a) act" Tr A /F(" ) = a- (aeA). In particular, if A = M2(F), then from (1.9) and (1.11) we obtain (1.12) We easily see that xL = j • l xj l for x G M2(F) with j = 0 - 1 1 0 , from which we easily see that t is indeed an F-linear involution of M2{F) satisfying (1.11). Now given an arbitrary quaternion algebra A and a G A, we define aL G A by aL — T r ^ / F ( a ) — a. This is an F-linear map of A into itself. Fixing an isomorphism a : A®F F — M2{F) and defining i on M2(F) by (1.11), we see that t is indeed an involution of A, since a(aL) = cr(a)L. Substituting a for t in (1.10), we obtain the first equality of (1.11).

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2004 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.