A.A LETTER FROM M. ROST TO J-P. SERRE 95
(which ma y b e calle d "reduce d n-fol d power" ) suc h tha t
(A.l) T n(£i{aubi}) = E
n
- ^ K
5
^ i r - - , ^ , ^
n
} -
To defin e T
n
tak e (A.l ) a s a definitio n o f T
n
o n forma l sum s o f symbol s {a , b} G
F* x F * an d chec k T
n
(x + {a , be}) = T
n
(x + {a , b} + {a , c}) etc .
Now note tha t
n
Tn\
X^
a
*'^})
=
{ai,^i,...,a
n
,^
n
}
2 = 1
while th e trac e for m o f A = (ai , 61) 0 (g) (an, &n) is ((ai , &2 an, &n))-
Hence th e T
n
ar e (cohomological ) operation s whic h generaliz e th e trac e for m
construction fo r product s o f quaternion algebras .
IV) Interpretatio n o f (3(p) fo r dime/ ? = 3 , 4, 5
I notice d th e following :
LEMMA.
(1) If dim (p = 3 and 5(p) = —1, then
f3(-ip) = 0^1 eD((p).
(2) If dim tp = 4 and S((p) 1, then
/?(-p) = 0 o l 6 % ) .
(3) J/dim ^ = 5 and 5(p) = 1,£/&e n £/ie following are equivalent:
(a) D(y» ) n [Z% ) / % )] ^ 0 .
(b) /?(-¥ ) = 0 .
(c) Z% ) c X%)Z%) .
Clearly a normalizatio n o f 5(p) fo r od d dimc p is no restriction a t all .
PROOF O F
(3) : Le t a G D(ip). The n
ip~ ( a i , . . . , a
5
)
with ai = a. Moreover ,
P(-P) = {l,-auaiaj)\ 1 z j 5
and
-0 = a ^ _L—£
is a subfor m o f f3(—ip) o f dimensio n 10 \ dim/?(—/?) . Henc e (3(—p) = 0 ^ 0 i s
isotropic^ a G D(p)D(ip).
[ . . . ]
M. Ros t
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