30 M . L. RACINE

(

o Q

Q 0

which is a maximal order of K0 by Lemma 3. By Lemma 2, P is a maximal

2 ' n

order of K_ . Hence, by Proposition 2, M = % n © is maximal. Denote J D

2n * n

by © ; by Lemma 3 any maximal order of K is isomorphic to P for some

o -ideal Q .

PROPOSITION 4. Let C = K the split quaternions, # = M(C ). Then

any maximal order M of J can be written M = J n E(L) where

n

L = 0 ) (ox. 0 ay.), a an o-ideal and {x.,y. } a symplectic bas e of v

i=l

a 2n dimensional K vector spac e (on which we let C act). M is

/ 0 Q

isomorphic to J n O where © =

° n

, a maximal order of C. Denote

V o o

this M by M ; M = M if and only if 0 = © (which by Lemma 5 is

Q Q Q Q Q

1 2 1 2

if and only if the clas s of a = the clas s of a in Q/0 ). Therefore the

isomorphism c l a s s e s of maximal orders of ? are in one-to-on e correspondence

2

with elements of the group Q/Q , where Q is the clas s group of K.

PROOF. Only the result s on isomorphisms remain to be shown.

Clearly if two maximal orders of C, P and .0 are isomorphic then

n n

£ n © = ^ n © Let L = 0 7 (ox. 0 a y . ) , L = 0 J (ov. 0 a w . ) ,

II 11 J. —' 1 A ' 1 £ r

1

1

Cd

1

(

x

. , y . } , {v.,w. } symplectic b a s e s of V, M, = ^ n E(L.) i = 1,2. An

isomorphism of M onto M induces an automorphism of #= KM = KM

which by Martindale's Theorem extends to the algebra with involution

(C , J ) = (K , J ). Such automorphisms are of the form

cp : X - A^XA, X, A

6

C AA 2 = 7I, 7

€

K ([16], p . 248) . Thu s M p = M .

But M p = $p () E(L A) = ^ n E(L A).