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).
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