48 M. L. RACINE

a

r

, -. € M„ if and only if a

€

fl. n ^ " v ( c ) . Similarly a

r

, € M if and only

[ 1 1 J 2 U

\LL\

L

/ d a 0 \ / o cb \

If a c *

n

n ? "

V ( C )

. If a. b c L n ? "

v ( c )

, then [ *

0 ° \ c a 0/ \0 d

2

b /

/ 0 d a c b \

_ I € M* . Since we assum e that v(c) is even when & ha s no

\ 0 cacb /

symmetric prime we may assum e that v(b) = -v(c). Let a = c d c

Since d = & , a € & ; v(a) = v(d ) - 2v(c) -v(c), therefore a

€

fl n ?" V

/O cb \ / o d a c b \ /0 cb - d acb \

and I - _ =

€

M' Since v(d ) v(c),

\ 0 d

2

b / \ 0 cacb ' \0 0 /

v (cb - d acb) = v(cb) = 0 and M' contains ue for some unit u of 0.

Similarly ve e M' for some unit v of ©. Therefore uve , vue c M'

/d c \ / 0 a \ / ca d a\

Now _ _ = _

e

M if and only if a

€

j f V ( .

\ c d

2

/ \ a 0/ y d

2

a ca J

Multiplying by uve and vue we obtain ^e ,Ce c M' Also

/ 0 cb \

Oe = Ce c M' Similarly Te C M ' Therefore

11

yo d2by

1Z

^

Z1 L

M c C c M' and we must have M' = C .

q. e. d.

The cas e v(c) odd is more complicated when $ ha s no symmetric prime and

will not be needed.

' «2 °

0 ) with d. € fl ,