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
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