28
M. L. RACINE
The proof of Lemma 6 yields that the number of isomorphism c l a s s e s of maxi -
mal orders of K is the number of equivalence c l a s s e s of the following
n
identification on the group Q/„n where Q is the clas s group: g ~ g
Q
Collecting results we have the following proposition.
PROPOSITION 3. Let C be the split field over K, £ = M(C ) s K .
' ° n ' n
Maximal orders of 9 coincide with maximal orders of K and the
n
isomorphism c l a s s e s of maximal orders of ? are in one to one correspondence
with equivalence c l a s s e s of the group Q/Q under the further identification
-1
g ~ g .
REMARK 1. If | Q/Q | is finite the number of isomorphism c l a s s e s of
maximal orders of # is the number of elements of Q/Q satisfying x = 1
1
plus 1/2 the number of elements satisfying x ^ 1.
Let C = K , the split quaternions. We need the following lemma.
LEMMA 7 ([44], p. 35). Let V be a vector spac e of dimension 2n
over K and g(x, y) a non-degenerat e alternating form on V. Let L be an
o-lattice of lr. Then there exist s a bas e {x, , . . . , x , y , , . . . , y } of V/K
1 n 1 n
and (fractional) o-ideals a , . . . , a such that g(x., x.) = g(y., y.) = 0,
g(x,,y. ) = 5 . . (in other words a symplectic base ) and
L = ox. 0 Q.y. 0 ox
0
0 Q_y. 0 . . . 0 ox 0 a y ; a, D ci D . . . D a and
i l l L L C . n n n l ^ , n
thes e ideals are uniquely determined by L and g.
Let p=U(C , J ) =U(C) ^U(K.J) where 0 = ( ? \), that is ,
n -y n ^dn O \-l 0/
the symplectic involution. Maximal orders of ^ are of the form E(L) n ?,
L an o-lattice of y a 2n-dimensional K-vectorspace. To determine which
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