REAL HILBERTIANITY-TOTALLY REAL NUMBERS
5
Applying Proposition 1.2 to
V =
A I yields the Block Approximation Lemma of
[P3, p. 354]:
COROLLARY 1.3. Let K be a PRC field. Let HI, ... ,
Hm
be disjoint clopen
subsets of X(K), and let
XI, ...
,Xm
E
K, and
VI, ... ,
Vm
E
Kx. Then there is
x
E
K such that for every
j
(x -
Xj )
2
~P
vJ for every P
E
Hi.
DEFINITION 1.4. Let K be algebraically closed with
t
E Aut(K) of order 2.
(a) ForcE
K
let lclz
= c ·
t(c).
(b) For z E j(n let llzllz
=
L:~=IIziiZ·
(c) For a K-linear morphism
f:
An
--+
Am given by a matrix A
=
(aij)
E
Mmxn(K) let llfllz
=
L:i,j laiiiZ·
In the above definition, the fixed field R oft is real closed, and llclz, llzllz, llfllz
are nonnegative elements
~f
R.
If
z ERn then llzllz
=
llzll
2
.
Also, in the unique
ordering of R, for all z E Kn the Schwartz inequality gives
(3)
llf(z)ll~ ~ 11!11~ ·llzll~·
REMARK 1.5. The space X('Qtr) is homeomorphic to Xw
=
{0,
l}No,
the
universal Boolean space of weight
N0
(cf. the concluding Remark of [FV3]). In
particular it has no isolated points.
LEMMA 1.6.
If
K is a finitely generated field, then the set xa(K) of archi-
median orderings on K is dense in X ( K).
PROOF. By induction on the number of generators of K/Q it suffices to show
the following. Let K/ K
0
be a simple extension of countable fields, let P E X(K),
and let Po
=
resK0 P E X(Ko).
If
Po is in the closure of xa(Ko), then Pis in
the closure of xa ( K).
The restriction X(K)--+ X(K0
)
is open
[ELW,
4.bis], hence we may assume
that Po is archimedian.
If
K/Ko is algebraic, then Pis also archimedian. Oth-
erwise
K
is the field of rational functions in one variable
t
over
K
0
.
Replace
(K0
,
P0
)
by its real closure (cf. [C, Lemma 8]) to assume that
K
0
is real closed.
By [C, Corollary 9(c)], every neighborhood
U
of
P
in X(K) contains a set
of the form {Q E X(K)I a
t
bin Q}, where a,b E K
0
and a bin P0
.
As Po is archimedian, we can embed K0 into
R
Since K
0
is countable, there is
c E JR'-Ko in the interval (a, b) in JR. This cis then transcendental over K
0
.
The
K
0
-embedding
K
--+
1R given by
t
~----+
c
induces an archimedian ordering Q on
K,
and
a
t
b in Q. Thus Q E U. o
For a subset I of a group G let Cone(!)
=
UaEe !"". We say that I is a
conjugacy domain,
if I is closed under the conjugation, that is, I= Cone(!).
DEFINITION 1. 7. Let
F / E
be a Galois extension of fields with
F
not formally
real. We say an involution
f
E G(F/E) is
real
if its fixed field F(E) in F is
formally real. Equivalently,
f
is the restriction of an involution in the absolute
Galois group G(E) of E. Let I(F/ E) be the set of real involutions of G(F/ E).
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