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).