MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
for every involution
if and only if
Assume (a). Then (b) means the conjunction of the following two statements
[FJ, Remark 25.14]:
Z) has a root c EM; and
has no root in
Assume (a) and (b), and let
E H be an involution. Condition (c) says
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
The following collection of conditions on a field M is
a primitive recursive set of elementary sentences in
= (JH) }.
(4) G(M) has the embedding property.
is totally real.
PROOF. For (1) see [P2, Theorem 4.1]. Condition (2) says each irreducible
E Q[X] has a root in M if an only if
has a root in Qtr. This is
splits over the real closure of Q. Express the latter in .C(Q) either
by Tarski's principle
Proposition 1.4] or by Sturm's Theorem
XI, §2]. Conditions (3) and (4) easily follow from Lemma 10.2.
Assume (1) and (2). By Remark 1.8(b), the image
X(Qtr) is closed in X(Qtr). Thus, (5) is equivalent to
dense in X(Qtr). Now,
X(Qtr) has a basis from the sets
extends to Qtr(o:)}, where o:
runs through the elements of Q. Indeed, by Remark 1.8(b) these sets are clopen.
Corollary 9.2], Qtr is SAP. That is, the sets
E X(Qtr)l c E P}
extends to Qtr(y'c)}
form a basis for the Harrison topology on
Qtr), as c varies on Qtr.
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
A field M is a model
if and only if it satisfies
PROOF. The conditions hold for
Qtr. By Remark 10.4 they thus hold
for each model
of Th(Qtr). Conversely, assume (1)-(5). Then
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