REAL HILBERTIANITY-TOTALLY REAL NUMBERS 3
that led to an improved presentation of this paper.
F. Pop told us the characterization of
G(Qtr)
also follows from his "!Riemann
existence theorem." His method uses rigid analytic geometry, versus our use of
the classical Riemann existence theorem. We look forward to seeing a written
account.
This paper corresponds to a portion of the talk of the first author at the Tempe
conference on Arithmetic Geometry, March 1993. Appropos the theme of the
Tempe conference, this paper uses profinite ideas of lwasawa for characterizing
absolute Galois groups of fields. The remainder of the talk discussed regular
realizations of dihedral groups. It considered the dihedral groups
Dpn
of order
2pn with pan odd prime and realization of these as Galois groups over
Q(x)
with
a bounded numbers of branch points [DF2, section 5.2]. The talk emphasized
the relation with rational points on modular curves. Extension of these ideas
with a general group
G
replacing
Dp
will appear under the title Modular stacks
and the Inverse Galois Problem.
1. Ordered fields and an approximation theorem for PRC fields
Let
K
be a field of characteristic 0, and let
G(K)
be its absolute Galois group.
Recall [Pl, §6] that the set of orderings
X(K)
of
K
is a boolean topological space
in its natural Harrison topology. This topology is given by a subbase consisting
of sets of the form
H(c)
=
{P
E
X(K)I
c E
P},
forcE Kx. Here
P
denotes the
positive elements in an ordering.
By Artin-Schreier theory
[L,
XI,§2], the real closures of
K
(inside a fixed
algebraic closure
K
of
K)
are the fixed fields of the involutions in
G(K).
This
identifies the set
X
of real closures of
K
with a topological subspace of
G(K).
It
is a boolean space, since the set of involutions is closed in
G(K).
Observe that
H(z)
=
{R
E
XI
z
E R} is open in
X,
for each
z
E
i(x.
For each R E
X
let 1r(R) be the restriction of the unique ordering of R
to
K.
Then 1r(R1
)
=
1r(R2
)
if and only if R
1
and R
2
are conjugate by an
automorphism of Kover
K.
The map'1r:
X---
X(K)
is continuous: 1r-
1
(H(c))
=
H (
ye).
Moreover, there exists a closed subset
X
of
X
such that 1r:
X
---
X ( K)
is a homeomorphism [HJl, Corollary 9.2]. The corresponding closed subset of
involutions in
G(K)
contains exactly one representative from each conjugacy
class of involutions. Having fixed such X, identify
X(K)
with it.
REMARK
1.1.
Comments on orderings.
(a) If
K
is PRC, then every clopen subset of
X(K)
is of the form
H(c)
for a
suitable c E Kx [P2, Proposition 1.3].
(b) Let R be a real closed field, and let a E R, c E Rx. If either c 0 or
a
0, then the system Y
2
+
cZ2
=a,
y
=J
0 has a solution in
R.
(c) For
X=
(XI, ...
'Xn) put
IIXII
2
=
L~l
x;.
Let K be an ordered field,
and let a, b, c E
Kn
and v E
Kx.
From the triangle inequality (over the
real closure of
K),
if
II
a- bll
2
,
lib- cll
2
(
~
)2
then
II
a- cll
2
v2
.
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