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
=
(!Hl·
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
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