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.