30. APPLICATION S 73
P R O P O S I T I ON 30.9 . q' = -(1,2 ) + (1,1,2)q - (1,2)X2q + X 3q
R E M A R K S 30.10. (1) A s in 30.6 , if we assume tha t k contain s th e 8t h root s o f
unity, 30. 9 simplifies as :
(30.9/) q' = q + \ 3
q
mW(k).
(2) B y Theore m 31.26, q can be writte n a s (1, 2) 0 Q wher e Q i s a quadrati c
form o f rank 4 . Similarly , q' ca n be written a s (1, 2) 0 Q'. Henc e 30. 9 can
be rewritte n i n term s o f Q an d Q' . Th e resul t i s muc h simpler , namely :
(30.11) Q' X3Q = (2d)Q, wher e d d(q) i s the discriminan t o f q.
30.12. A N O T H E R EXAMPLE . Conside r th e subset s o f { l , . . . , n } wit h tw o
elements. Th e actio n o f S
n
o n thes e subset s give s a homomorphis m
ip: S
n
SN wher e N =
(This homomorphis m i s onl y define d u p t o conjugation. ) A s above , thi s give s a
map Et
n
Et_/v; cal l E(2) th e etal e algebr a o f ran k N correspondin g t o a n E i n
Et
n
(/c). (I n classical terms , thi s correspond s t o taking th e sums xi -\-Xj o f the root s
of a n equation , provide d th e equatio n i s genera l enough . Scheme-theoretically , i f
X i s a finite etal e coverin g o f 5 , on e define s X(2) b y removin g th e diagona l fro m
X Xs X an d dividin g b y th e involutio n (x,y) ^ (y,x).) B y §29 , one know s tha t
the trac e for m qE(2) c a n b e computed i n terms o f the exterior power s A 2(g#). Mor e
precisely:
P R O P O S I T I O N 30.13. One has the formula:
QE(2) = A 2 (te) - (2 ) (qE - n).
P R O O F . Th e formul a hold s whe n r a n k E = 1 (wher e E(2) = 0).
When rank^ E = 2 , w e hav e E(2) = & , q
E
{2) 1? a n d qE = (2,2a ) fo r som e
a G k*. The n \ 2(qE) = (OL) and
(2)(qE-n) = (l,a)-(2
J
2) = (a)-l,
since (2 , 2) = (1,1) - Henc e th e formul a hold s i n thi s case .
For E = E' x E"\ on e has the formul a
E{2) 9 * E'{2) x (E' 0 E") x E"(2),
which implie s a correspondin g equalit y fo r trac e forms :
QE(2) = QE(2) ®qE'qE © t e - ( 2 ) -
This show s b y induction o n n tha t th e proposition hold s whe n E i s multiquadratic .
By Th . 29.1, i t hold s i n general .
R E M A R K S 30.14. (1) Th e formul a fo r th e trac e for m o f E(3) (define d i n a n
obvious way ) is
qBi3)=\3(q)-2-X2(q) + n-(2)iq-l),
GO-
where q = q
E
an d n = r a n k ^ .
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