ARITHMETICS OF JORDAN ALGEBRAS ?

(7) N ( l ) = l , N(aU ) = N(a) N(b) 2 for all a, b

6

% .

(8) An element a

€

^ is invertible if and only if N(a) £ 0 .

(c) (c)

McCrimmon [34] has shown that the generic norm N of the isotope g

can be expressed in terms of the original generic norm N by the formula

(9) N ( C ) = N(c) N .

Define T(a, b) = - A ^ A a log N = (A a N)(A ^ N ) - A ^ A S N where A a N is

X X i A J. X X

the directional derivative of N in the direction a, evaluated at x, i . e . the

coefficient of 7\ in N(x + 7\a), a, b € #, x a variable in p. Jacobson and

Katz [1 8] have shown that A a N = T(a) = T(a, 1). We wish to show that

A j A ^ N = a

2

(a,b ) = a

2

(a + b) - a

2

(a ) - c

2

(b). We have A a N = the

coefficient of ~k in N(x + 7\a) = coefficient of in 7\ (N(A x + a)) = the

coefficient of " A in N(Ax + a); since 7v(l + ^b) + a = 7\1 - (-(a + H-b)),

we have A , A N = the coefficient of |J. in (the coefficient of 1\ in

1 x

(7\ - 7 v cr (-(a + fyib)) + A cr

2

(-(a+ Ay.b))-...) ) = the coefficient of jj. in

2

(a (a) + cr (a,b)|jL + terms involving JJ. ) = cr (a, b). Therefore we have

(10) T(a

f

b) = T ( a ) T ( b ) - a

2

( a , b )

Let $ be a vector spac e over the field $, Q a quadratic form on

P with bas e point 1 ( i . e . Q(l ) = 1). Let x, y

€

$, then

(11) yU = Q ( x , y ) x - Q ( x ) y , where