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) =
cn
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 lxj 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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