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