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
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