REAL HILBERTIANITY-TOTALLY REAL NUMBERS

27

If

there is an epimorphism

1r:

H

--

A,

let

H

=

H.

If

not, let

H

=

(H

x

A,IH

x {a}). Both

A

and Hare quotients of

H.

Observe that(*) holds for

H.

Otherwise

IH

X

{a} generates a proper subgroup

r

of H

X

A such that the

projection

H

x

A-- H

maps

r

onto

H

=

(!HlÂ·

Thus

r =

{(h, 1r(h))l hE

H},

where

1r: H-- A

is an epimorphism. As

IH

x {a}

~

r,

we have

1r(IH) =a.

Thus

1r

induces an epimorphism H

--

A,

a contradiction.

PROOF OF (c). Clear from (a) and (b). o

If,

in addition to the assumptions of the theorem,

K

is countable, then

G

is

of rank at most N0

.

Thus the involutory structures G and D are very similar,

by Lemma 7.3 and Theorem 7.4. In fact, we have the following straightforward

modification of

[F

J,

Lemma 24.1

J.

LEMMA 7.5.

Let

G

and H be involutory structures with embedding property,

such that G and H are

of

rank

at

most

N0

.

JflmG = ImH,

then

G

~H.

THEOREM 7.6.

Let K be

a

countable totally real Hilbertian PRC field. As-

sume that K has no proper totally real algebraic extensions and

X (

K) has no

isolated points. Then G(K)

~

D,

and hence G(K)

~D.

The field Qtr of totally real algebraic numbers is PRC by

[P].

(We remark

that although Pop

[P]

states this result, he only gives the proof for an analog.

Therefore in all our results about Qtr the reference

[P]

should be replaced by a

subsequent version, where this omission will be remedied.)

It

is clearly countable.

By Proposition 6.6 it is totally real Hilbertian, and by Remark 1.5, X(Qtr) has

no isolated points. Therefore:

COROLLARY 7. 7.

The absolute Galois group

of

the field

Qtr of

totally real

algebraic numbers is the free profinite product D

of

groups

of

order 2 over the

universal Boolean space Xw

=

{0,

1}No of

weight N0.

8. Real Frobenius fields

Let

SIR

be a real Galois ring cover, and let

FIE

be the corresponding field

extension. Let

K

be a subfield of Rand

L

the algebraic closure of

Kin F.

The following definitions are valuable

[HL,

Definition 4.2].

(a)

SIR

is

regular over K,

if the extension

E I K

is regular. In that case

L

I

K

is a finite Galois extension.

(b)

SIR

is

finitely generated

over

K,

if

R

and

S

are finitely generated

rings over

K.

(c)

FIE

is

amply real

over

K

if

E I K

is a regular extension, the algebraic

closure

L

of

Kin F

is not formally real, and the extension

F(E)IL(E)

is

totally real for every real involution

E

E

G(F

I

E).

DEFINITION 8.1. A field M is said to be

real Frobenius

if it satisfies the

following condition: Let

Sl

R

be a real Galois ring cover, finitely generated and

regular over

M,

with

FIE

the corresponding field extension amply real over