REAL HILBERTIANITY-TOTALLY REAL NUMBERS

23

[FJ,

Lemma 5.5]. If 'lj;' is another extension of¢, then '1/J'* and 'lj;* differ by an

inner automorphism of G(F/E)

[L,

Corollary 1 on p. 247]. In particular, for

a E G(M) and a conjugacy domain

I~

G(F/E) we have 'lj;'*(a) E I if and only

if 'lj;*(a) E I. This allows us to abuse the notation and write ¢*(a) E I instead

of 'lj;*(a) E I. (Cf. also

[HL,

Remark 4.1].)

REMARK 6.2. Let S

0

/

Robe another real Galois cover ofrings, with L/ K the

corresponding extension of the quotient fields. Assume that

R

is finitely gener-

ated over Ro, the field

K

is algebraically closed in

E,

and

L

is the algebraic clo-

sure of Kin F. Furthermore, Ro

~

M and¢: R-- M is an flo-homomorphism.

We may choose the extension 'ljJ of¢ to be an So-homomorphism. From (1):

(a) Composition of 'lj;* with res£: G(F /E)

--

G(L/ K) is the restriction map

res£: G(M)

--

G(L/ K).

(b) ForeachP E X(K), ¢*(Ip(M))

~

Ip(F/E). Thus, ¢*(I(M))

~

I(F/E).

Indeed, let E E Ip(M), and let 'lj;:

S

--

M

be an extension of ¢. It follows

from (1) that 'ljJ maps the integral closureS' of

R

in F('lj;*(E)) into M(E). The

latter field is real closed. Thus, by Knebusch's Theorem

[HL,

Proposition 1.2],

P

extends to an ordering on F('lj;*(E)).

Through this section and for an ambient field

K

consider the following setup:

xis transcendental over K, E = K(x) and R = K[x, h(x)-

1].

Also, Sis a real

Galois cover of R with F / E the corresponding extension of quotient fields.

(2) Let X

1 , ... ,

Xm be a partition of X(K) into disjoint clopen subsets. For

each 1

:S

j

:S

m, let Q1 E X(E) so that resKQj E

.x1.

Therefore, a E K with h(a)

"#

0 defines ¢a: R-- K by x ,_...a.

LEMMA 6.3. In

(2)

let

Q

be an ordering on E with

P

its restriction to K.

Denote the real closure of (K,

P)

by K. There exist branch points x

1

x

2

in K

of the extension F / E with no other K branch points between them. They have

this property. For each a

E

Kin the interval

(x1,x2),

¢~(Ip(K)) ~

IQ(F/E).

PROOF. Let E E IQ(F/E). Choose a primitive element y for F(E)/E, integral

over K[x]. Let

J,

= irr(y,

E)

E K[x, Y] and

S'

the integral closure of

R

in F(E).

The sentence (:JX, Y)[j,(X, Y) =

OA U(X,

Y)-# 0] holds in F(E), and therefore

also in the real closure of (E,

Q).

By Tarski's principle it is valid in

K.

Thus

there is

a

E K such that

J,(a,

Y) has a simple root inK. This certainly remains

true if

a

is replaced by a in the neighborhood U of

a

in K determined by nearest

branch points of

F / E

in

K.

Let

L

be the generator of G(K), and let

S'

be the integral closure of

R

in

F(E). Let

a

E U

n

K. Then ¢a: R

--

K

~

K extends to a homomorphism

'1/J: S'-- K. It follows that its extension 'lj;: S--

K

satisfies '1/J*(t) =E. As Ip(K)

and IQ(F/E) are the conjugacy classes of LandE in the respective groups, we

have

¢~(Ip(K)) ~

IQ(F/ E). o

DEFINITION 6.4. A formally real field K is

totally real Hilbertian,

if in

each setup (2) the following holds. When G(F/E) =

(U,T=

1

IQ1 (F/E)), then