5. VERSA L OBJECT S
13
(2) G Sn, s o A = Et
n
. Tak e K =
fco(ci7
cn) wher e th e C{ ar e indeter -
minates. Defin e th e i^-etal e algebra :
E = K[x\/{x
n
+ dx
71'1+
+ c n).
The correspondin g torso r i s versal; i t i s the specia l cas e of 5. 5 wher e V i s
affine n-spac e an d S
n
act s b y permutin g th e coordinates , se e 24.6.
5.7.
ESSENTIA L DIMENSION .
Th e minima l transcendenc e degre e o f K/k
0
fo r
K th e field of definition o f a versal G-torso r i n 5. 1 is called th e essential dimension
of G, cf . [R e 00]. Thi s is an interestin g invarian t o f G
1
whose explicit determinatio n
is not eas y i n genera l (eve n i f G i s finite). Her e ar e som e example s (assumin g th e
characteristic i s ^ 2) :
ed(On) = n fo r n 1
ed(SOn) = n-l fo r n 3
ed(G2) = 3.
The cyclic group of order n ha s essential dimensio n 1 if ko contains a primitive n-t h
root o f unity . However , th e cycli c grou p o f orde r 4 has essentia l dimensio n 2 if ko
does not contai n a primitive 4th root o f
unity.e
Man y mor e examples may be foun d
in [R e 00].
REMARK
5.8 . On e coul d als o broade n th e notio n o f versal torso r b y removin g
the densit y conditio n i n 5.1.2; tha t is , by replacin g 5.1.2 wit h th e following :
(2;) Fo r ever y extensio n k o f & o with k infinit e an d ever y G-torso r T ove r k,
there exist s x £ X{k) whos e fiber Q
x
i s isomorphic t o T .
The result s her e specifically, 12.3 and 27.13 still hol d wit h thi s broade r defi -
nition.
eSee [Se92b , p . 6 , exercise] . Not e a misprin t i n thi s exercise : +6Z 2 shoul d twic e b e replace d
with - 6 Z
2
.
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