32
MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
(c)
'1/J*(I(M))
=
IH:
for every involution
f
E H,
f
E
IH
if and only if
(c)
f
E
'1/J*(J(M)).
Assume (a). Then (b) means the conjunction of the following two statements
[FJ, Remark 25.14]:
(b1) fH(a,g(a)-
1
,
Z) has a root c EM; and
(b2) if
G H,
then
fc(a,
g(a)-
1
,
Z)
has no root in
M.
Assume (a) and (b), and let
f
E H be an involution. Condition (c) says
(c~)
h(a,g(a)-l,c,Z') has a root o: EM such that M(o:) is formally real.
Therefore the assertion follows by Lemma 10.2. o
PROPOSITION 10.4.
The following collection of conditions on a field M is
equivalent
to
a primitive recursive set of elementary sentences in
.C(Q).
(1)
M
is PRC.
(2) MnQ
=
Qtr.
(3) ImG(M)
={HI
H
= (JH) }.
(4) G(M) has the embedding property.
(5) M/Qtr
is totally real.
PROOF. For (1) see [P2, Theorem 4.1]. Condition (2) says each irreducible
polynomial
f
E Q[X] has a root in M if an only if
f
has a root in Qtr. This is
equivalent to
f
splits over the real closure of Q. Express the latter in .C(Q) either
by Tarski's principle
[HL,
Proposition 1.4] or by Sturm's Theorem
[L,
Chapter
XI, §2]. Conditions (3) and (4) easily follow from Lemma 10.2.
Assume (1) and (2). By Remark 1.8(b), the image
X
of restriction
X(M)
---
X(Qtr) is closed in X(Qtr). Thus, (5) is equivalent to
X
dense in X(Qtr). Now,
X(Qtr) has a basis from the sets
{P
E X(Qtr)l
P
extends to Qtr(o:)}, where o:
runs through the elements of Q. Indeed, by Remark 1.8(b) these sets are clopen.
By
[Pl,
Corollary 9.2], Qtr is SAP. That is, the sets
H(c)
=
{P
E X(Qtr)l c E P}
=
{P
E X(Qtr)l
P
extends to Qtr(y'c)}
form a basis for the Harrison topology on
X (
Qtr), as c varies on Qtr.
It
suffices to consider only o: E Q with Q( o:) formally real; otherwise the
corresponding set of orderings is empty. Thus (5) is equivalent to this statement:
If Q(o:) is a finite formally real extension of Q, then M(o:) is formally real. Now
use Lemma 10.2. o
COROLLARY 10.5.
A field M is a model
ofTh(Qtr)
if and only if it satisfies
conditions (1)-(5).
PROOF. The conditions hold for
M
=
Qtr. By Remark 10.4 they thus hold
for each model
M
of Th(Qtr). Conversely, assume (1)-(5). Then
M
is a real
Frobenius field (Proposition 8.3), and ImG(M)
=
ImG(Qtr), by (3). So, (2) and
(5) imply res!QG(M)
=
G(Qtr). By Proposition 9.4 (with K
=
Q) the fields M
and Qtr satisfy the same sentences in .C(Q). o
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