REAL HILBERTIANITY-TOTALLY REAL NUMBERS
31
LnQtr, then Qtr
j K'
is totally real, and hence resK'X(Qtr) =
X(K').
Therefore
resLG(Qtr) =
(G(L/K'),I(L/K'))
=
G(LjK').
Let{} be a sentence in .C(Q). Proposition 9.4 effectively gives a finite Galois
extension L of Q with
A
E L, a finite family H of (isomorphism types of)
finite involutory structures, and, for
S
={HE
HI IH-=/- 0
is a conjugacy domain in Hand
H
=
(IH)},
a conjugacy domain Con = Con(S) in Sub[L/Q] contained in
S.
For these,
Qtr
f= {}
if and only if
G(L/ L
n Qtr) E Con. This condition is checkable, by the
remarks preceding this corollary. o
LEMMA
10.2. There is
a
formula
O(X1
, ... ,
Xn)
E .C(Q)
with the following
property. Let M be
a
PRC field, let
a
=
(a1
, ... ,
an)
E
Mn,
and
put
f =
zn
+
a1 zn-
1
+···+an
E
M[Z].
Then M
f=
O(a)
if
and
only if
( *)
f
has
a
root a in M such that M (a) is formally real.
PROOF. Condition (
*)
is equivalent to this: There is an ordering P on M
such that
f
has a root in the real closure of
(M, P).
By Tarski's principle
[HL,
Proposition 1.4] this is equivalent to a finite disjunction of statements of this
form: There is an ordering
P
on
M
with /\~= 1 fi(a) = 0
1\
/\';=1
gj(a)
E
P,
where
/I, ...
Jng1, ... ,gm E Z[X1, ... ,Xn] do not depend on Manda.
Put
g0
= 1, and let
D.
be the set of finite sums of squares in M. By
[Pl,
Corollary 1.6],
I::J=o
gi(a)gj(a)D.
is the intersection of all orderings on
M that contain g1(a), ... , gm (a). Therefore the last statement is equivalent to:
/\~= 1 /i(a) = 0
1\
-1
~
I::J=o
gi(a)gj(a)D..
As
D.
is the set of sums of two
squares in the PRC field M
[P2,
Proposition 1.5], a formula in .C(Q) expresses
this statement. o
PROPOSITION 10.3.
Every real Galois formula{} over
a
field K is equivalent
to a formula in .C(K), modulo the theory of PRC fields M containing K.
PROOF. It suffices to express in
.C(K)
the statement
Ar(A,
X) E Con, with
C/A
a real Galois ring/set cover over
K
and Con= {HalO" E
G(C/A)},
where
H =
(H,IH)
is a involutory substructure of
G(C/A).
Let
E
=
K(A)
and
F
=
K(C)
be the quotient fields. For each G::;
G(C/A)
=
G(F/E)
let
F(G)
be
the fixed field of
G
in
F,
and let
zc
be a primitive element for
F (G) j E.
Replacing
A
by an open subset
A'
(that is, replacing the given Galois stratification by its
refinement) we may assume that K[A][zc]/K[A] is a ring cover
[FJ,
Definition
5.4] and
zc
is a primitive element for it.
Write K[A] as K[x,g(x)-
1],
where xis a generic point of
A
over
K.
Let
fc
be
a polynomial over K such that
fc(x,
g(x)-
1,
Z) = irr(zc, E). For every
E
E H
let
h,
be a polynomial over
K
with
h,(x,g(x)-\zH,Z')
= irr(z(,),F(H)).
Then
M
f=
Ar(A,
a) E Con means the following conditions hold.
(a) a E A:there is a specialization x---- a such that g(a)
-=/-
0.
(b) x---- a extends to a homomorphism
'1/J:
C ____, M such that 'lj;*(G(M)) =H.
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