REAL HILBERTIANITY-TOTALLY REAL NUMBERS

25

set of involutions in

G.

A morphism of involutory structures ¢: G ---+ H is a

continuous homomorphism of groups¢:

G---+

H such that

¢(Ic)

~

IH.

We say

that¢: G---+ His an epimorphism if ¢(G)= Hand

¢(Ic)

=

IH.

EXAMPLE 7.1 (A). Let L

I

K be a Galois extension with L not formally real.

Then

G(LIK)

=

(G(LIK),I(LIK))

is an involutory structure. Let

E

be an

extension of

K,

and let

FIE

be a Galois extension such that

L

~

F.

Then the

restriction resL:

G(FIE)

---+

G(LIK)

is a morphism. Moreover, suppose that

E

I

K

is regular and totally real: every ordering on

K

extends to

E.

Then resL

is an epimorphism

(cf.

[HJl, Lemma 3.5]).

EXAMPLE 7.1

(B).

Let

SIR

be a real Galois cover with

FIE

the correspond-

ing Galois extension of fields. Let

M

be a field and let}!_:

R

---+

M

be a ho-

momorphism. Extend ¢ to a homomorphism 'lj;:

S

---+

M.

Then the group

homomorphism 'lj;*:

G(M)

---+

G(F I E)

is a morphism of involutory structures

'lj;*: G(M) ---+

G(FI E)

(Remark 6.2(b)).

A finite image of G is a finite involutory structure H for which there exists

an epimorphism ¢: G ---+ H. Clearly, up to an isomorphism, it is of the form

(GIN, {ENINI

E E

Ic}),

where

N

is an open normal subgroup of

G

not meeting

Ic.

Let ImG be the class of all finite images of G.

A finite embedding problem for G consists of an epimorphism

Jr:

H---+

A

of finite involutory structures, together with an epimorphism ¢: G ---+ A. A

solution is an epimorphism 'lj;: G ---+ H such that

1l'

o 'lj; = ¢. We say that

G has the embedding property if every finite embedding problem (

Jr:

H ---+

A,

¢: G ---+

A)

for G, in which H is a finite image of G, has a solution.

EXAMPLE 7.2. Let

D

be the free profinite product

fixEXw

(Ex) of groups of

order 2 over

Xw

(Remark 1.5), and let

Ifjy

= {Exl x E

Xw}·

A finite involutory

structure (A,

IA)

is a finite image of (D,

Ifjy)

if and only if A is generated by

IA.

Furthermore,

(D, Ifjy)

has the embedding property.

With

D

and

Ifjy

as above, put D =

(D, ID),

where

ID

is the conjugacy domain

ConD(Ifjy) of D generated by

Ifjy.

LEMMA 7.3. (a)

ID is all involutions in D, and Dis of rank

~ 0 .

(b)

A finite involutory structure

A

is in

ImD

if and only

if

IA

=1-

0

is a

conjugacy domain in A and A= (IA)·

(c)

D

has the embedding property.

PROOF OF (a). See [HJ2, Corollary 3.2 and Lemma 2.2]

PROOF OF (b). Immediate from Example 7.2.

PROOF OF (c). Let

Jr:

H---+

A,

¢: D---+

A

be a finite embedding problem for

D. Then

IA

~

A and

IH

~

Hare conjugacy domains. Let

I~

=

¢(Ifjy)

and let

I~

= {E E IHJK(E) E

I~}.

As ConD(Ifjy) =

ID,

we have

ConA(J~)

=

IA;

it

follows that Con

H

(I~)

=

I

H.