ARITHMETICS OF JORDAN ALGEBRAS 25
0 81
D . . . JD
si"1
. . .
aT1
o'
where $v is the right order of 81 and the "matrix" has the same meaning as
in Lemma 2.
LEMMA 4. ([5], p. 16) Let L. i = l , 2 be ©-lattice s of U. Then
E(L ) = E(L ) if and only if L ^ 81L for some (two-sided fractional) O-ideal
81; E(L ) = E(L2) if and only if L - 8IL
r
LEMMA 5. ([5], p. 21) If & = K two lattice s
L, = a x . 0 . . . 0 Q x , L = b v © . . . ® b y are isomorphic if and only
1 1 1 n n 2 1 1 n n
if the ideal a, a . . . a is in the same clas s a s the ideal b b . . . b .
1 2 n 1 2 n
LEMMA 6. If $ = K the isomorphism c l a s s e s of maximal orders of
$ are in one-to-on e correspondence with elements of the group Q/^n where
n Q
Q is the clas s group of #.
PROOF. By Lemma 4, E(L ) = E(L ) if and only if L s QL for some
ideal a. Writing L, as in Lemma 5, E(L.) = E(L ) if and only if
Q, a . . . Q is in the same clas s as obn ob
o
. . . ctb = a b b_ . . . b (& is
1 2 n 1 2 n l 2 n
commutative). The proof is complete once one notes that 0 is unique (it
equals o).
q. e. d.
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