26

M. L. RACINE

§3. Maximal orders of H(C , J ), n 3, C a split associative composition

n

7

algebra.

Let $ be a field. A quadratic form Q on a $ vector space 1/ is

said to be non-degenerate if Q(x) = 0 and Q(x, lr) = 0 imply x = 0. A

composition algebra C over the field $ is a unital $-algebra endowed with

a non-degenerate $-valued quadratic form n satisfying n(l) = 1 and

n(ab) = n(a)n(b). We refer to [3] for the determination of composition

algebras and their elementary properties. Denote by t(a,b) the trace form

n(a + b) - n(a) - n(b) and by the canonical involution a -* a = t(a)l - a

where t(a) = t(a, 1). In this chapter and the next we make the further assump-

tion that C is associative. The octonions are treated in Chapter IV.

Let 7 = diag{ y , . . . , 7 }, -y , e $ , 7 . + 0, n 3. Then

J : A -+ y A 7 is an involution of C if A = (a..) for A = (a,.) and

7 n

IJ IJ

A = the transpose of A. Let # = U(C , J ) the symmetric elements of the

n -y

algebra with involution (C , J ) whose diagonal entries belong to $. ^ is

n

7

a special central simple Jordan algebra of degree n over $. Moreover ^

generates C .

n

Assume from now on that the base field is K the quotient field of a

Dedekind ring 0 and that C is split, that is, C is not a division algebra.

In that case either C = K 0 K with the exchange involution (the split field

case) or C = K with