54 M. L. RACINE

We give next a few global results . Let C be a division composition

algebra over K the quotient field of a Dedekind ring o, the standard

involution of C. Let & = K, so d

e

& implies that d is central. Hence

d(&Q n ?" v ( d ) ) = (dK) n (dp~v(d)) = K n O = o and (d(&Q n ? " v ( d ) ) ) ' = o.

Denote by J the involution of C induced by diagfy , . . . , 7 }, 7 . h 0,

7 . e K. Since hermitian forms can be diagonalized any involution of C

induced by a hermitian form h satisfying h(v, v) e $

n

, V v e V is of this form.

Also if n = 2, tt(C , J ) is of degree 2, a cas e treated in I §3.

COROLLARY 4. Let C = K the quotient field of a Dedekind ring 0,

9 - M(C , J ), n 3. An order of M of ^ is maximal if and only if

M = Q n M ' and M' is a maximal J -stabl e order of C .

0 9 Jy n

PROOF. If M is maximal, localize and note that when C = K we have

O = o , so 0 + *p = ©. Hence (6) and (7) are of the form (5) and M' is a

maximal J -stabl e order. The converse is simply Proposition 2.

q. e. d.

COROLLARY 5. Let C be a quadratic field extension over K the

quotient field of a Dedekind ring 0. An order M of ^ = M(C , J ) n 3 is

maximal if and only if M = J n M' and M' + 01 is a maximal J -stabl e

order of C .

n

PROOF. Assume that M = # n M' and that M' + 01 is a maximal

J -stabl e order of C . Since C is commutative, O is the integral closure

of 0 in C and is unique. Moreover O can be embedded in C a s 01 in*

n

only one way. Now 01 n 9 - ol which belongs to any order. So if M' + 01