REAL HILBERTIANITY-TOTALLY REAL NUMBERS
21
Part 2. Reduce to commutators generate M(G), H
~Aut(
G), and IIi
I~
2.
As Lis not formally real, 1
~
lx;(LIK) = 1r(Ii), for each 1
~
j
~
m. Thus
Ii ~ H'-.G. Let
ii
and
G
be as in Lemma 3.2, and let
ij
be the inverse image of
Ij in the set of involutions of
ii.
Suppose there is
F
regular over L, with
F I
K(x)
is Galois, and an isomorphism h:G(FIK(x))--+
ii
that maps G(FIL(x)) onto
G
and lx;(FIK(x)) onto
ij.
As in Part 1, the subfield ofF corresponding to
the kernel of the map
ii
--+
H
(sending
G
to G) is the desired
F.
Thus, assume that commutators generate M(G), CH(G) = 1, and IIi
I~
2 for
each
j.
In particular, the conjugation action of HonG induces a monomorphism
H--+ Aut( G). Identify H with its image in Aut( G) (and G with Inn( G)). Then
G(LI K) is a subgroup of Out( G)= Aut(G)IInn(G), and
1r:
H--+ G(LI K) is the
restriction of the quotient map
1r:
Aut( G)--+ Out( G) to
H.
Part 3. Construction. Let A : 'H --+ 'R be the cover of Hurwitz spaces,
associated with G, defined in Section 4. Let {3: G(K) --+ Aut('HI'R) be the
composition of the restriction G(K)--+ G(LI K)
~Out(
G) with the isomorphism
8: Out( G)--+ Aut('HI'R) (2.14). Furthermore, let v be as in Remark 4.2(d).
Let
M
be the field generated over Q by
v'-1
and the conjugates of basic
points associated with G, t:, and I as in Definition 4.1, for all possible t: and
J.
This is a finite extension of Q (Remark 4.2(b)). Refine the partition X1, ... , Xm
of X(K), and hence also the corresponding partition X1, ... , Xm of X, so that
for each 1
~
j
~
m
there are unique f.i
E
G(LIK) and
lj E
G(MIQ) such that
resLXj
= {f.j} and resMXj =
{li}·
Fix 1
~
j
~
m. Put Ij = {t:
E
Ijl7r(t:) = f.j} and clroose
Ej
E
Ij. Then Ij
~
G4(t:j)· Let
q
be the basic point associated with G, Ej, and Ij. Then Q(q)
~
M.
As the real involutions in G(MIQ) are conjugate, there is
Aj
E
G(MIQ) with
Xj
1ljAj
= resMc: cis complex conjugation. Set Qj = Aj(q). By Lemma 4.3,
8,(q) = c(q). Therefore, 8,(Qj) = 8,..j(q)..j8,(q) =
(..jc..j1)(Qj)
=
lj(Qj).
Let
~ E
Xi. Then 8,; = 8(1r(t:i)) = 8(f.j) = (8
ores£)(~)
=
{3(~).
So
~(Qj)
=
lj(QJ)
=
8,;
(QJ)
=
{3(~)(Qj).
Thus Q1, ... , Qm satisfy (1). Therefore there
exists
p
E
'H(K) that satisfies (2)-(4). Let FIK(x) be the Galois extension and
h1: G(FI K(x))--+ Aut(G) the embedding associated with
p
over K (2.17).
Part 4. The image of h1. Let r
E
H. There is u
E
G(K) such that resLu =
1r(r). By (2), a(p) = 8(resLa)(p) = 8r(p). Hence by the criterion of (2.17), r is
in the image of h1. Thus
H:::;
im(h1). But
IHI = IGI·IG(LIK)I = [F: L(x)]· [L: K] = [F: K(x)] = lim(hl)l,
and hence
H
= im( h1).
Part 5. h1
Ux;
(F
I
K(x))) =
!1.
Let 1 :::;
j :::;
m
and
P
E
Xi. By Prop. 4.4( d),
h1(lp(FIK(x))) = ConH(lj). As Cona(K)(Xj) = lx;(KIK), Cona(L/K)(f.i) =
Ix;(LIK). Conclude: ConH(Ij) =
11.
Thus h1(lp(FIK(x))) = Ii. o
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