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 ,
Previous Page Next Page