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
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