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).