18. COHOMOLOGICA L INVARIANT S O F Pfister
n
AN D Oc t
43
EXERCISE 17.5. (1) Le t x , y G k* wit h x + y / 0 . Sho w tha t
(x) -(y) = (x + y) -(-xy) i n H 2(k)
by proving that th e quadratic form s (x , y) an d (x -\- y,xy(x + y)) ar e isomorphi c
and computin g thei r wi invariant.
(2) Le t x , y, z £ k*. Pu t
a = x + y + z , C 2 = #2 / + y z + zx, c
3
= xyz .
Prove th e followin g identitie s i n H 2(k) an d H 3(k):
(2a) (x ) -(y) + (y) -(z) + (2; ) -(x) = (d ) -(-c
3
) + (cic
2
- 9c
3
) -(-cic3 ) an d
(2b) (x ) -(y) -(z) = (ci ) -(c3) -(cic2 - 9c
3
)
under th e assumptio n tha t c\ ^ 0 andC1C27^ 9c3.
[Hint: Le t E = k x k x k, an d le t q b e th e diagona l quadrati c for m o n
E define d b y (x,y,z). Th e lef t sid e o f (2a ) i s W2(q). Defin e ei , e
2
G i 2 b y
ei (1,1,1) an d e
2
= (y z,z x,x y). Th e vector s and e
2
ar e orthogona l
and anisotropic . Choos e e
3
orthogona l t o ei , e
2
an d nonzero . I f on e compute s
u2(q) usin g th e basi s {ei,e
2
,e
3
}, on e finds th e righ t sid e o f (2a) . Identit y (2b )
is deduced fro m (2a ) b y multiplicatio n b y (c
3
) = (x ) + (y) + (z).]
(3) Le t k = ko(xi, ..., x
n
), wher e th e x% are independen t indeterminates , an d le t
ci, ... , c
n
b e th e elementar y symmetri c polynomial s i n th e x%. Sho w tha t
there exist s a quadrati c for m ove r ko(c\, ..., c
n
) whic h become s isomorphi c t o
(xi,... , x
n
) ove r k. Deduc e that an y symmetric polynomial in (xi), ... , (x
n
) ca n
be computed b y universal polynomia l formula s fro m th e symmetri c polynomial s
in th e x^'s , just a s abov e fo r n 3.
18. Cohomologica l invariant s o f Pfister
n
an d Oc t
We no w determin e Inv(Pfister
n
, Z/2Z) . A n n-Pfiste r for m ma y b e writte n a s
(f)= ( l , - a i 0 . . . 0 ( l , - a
n
) .
Then
is a n invarian t o f fi b y [E L 72, 3.2] .
T H E O R E M 18.1The . group Invfc
0
(Pfistern,Z/2Z) of cohomological invariants
of n-Pfister forms is a free H(ko)-module with basis { l , e
n
} .
P R O O F . Fo r simplicit y o f notation , w e giv e th e proo f onl y fo r th e cas e n 2.
A simila r argumen t work s fo r an y n.
Let (\) = (1, -a) 0 (1, -/?) G Pfister2(/c). An y a G Inv(Pfister2, H) give s b y re -
striction a n invariant o f the pai r (a, /?) , where a an d f3 are viewed a s one-dimensiona l
quadratic forms , o r a s element s o f /c*/&* 2 o r a s Z/2Z-torsors . B y Theore m 16.4 a
is o f th e for m
(18.2) a{t) = A
0
+ (a) -A i + (/? ) -A2 + (a ) -(/3 ) -A12
where th e A' s belon g t o H(ko). Sinc e a an d / ? ma y b e swappe d withou t changin g
0, w e hav e A i = A
2
. Also , (p = (1, —a) 0 (1, af3), s o
(18.3) a(t)) = A
0
+ (a ) -A i + (-a/? ) -A i + (a ) -(-a/3 ) -A
12
.
By addin g (18.2) an d (18.3) w e get (-a) -A i + (a) - ( - a ) -X
12
= 0 . Sinc e (a) -(-a) = 0 ,
we hav e (—a ) -A i = 0 . Thi s i s true fo r ever y extensio n k o f ho an d fo r ever y a G &*;
that is , th e invarian t
6 = ( - l ) - A i + i d - A i inInv
f c o
(Z/2Z,Z/2Z)
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