REAL HILBERTIANITY-TOTALLY REAL NUMBERS
19
for 1:::; j:::;
e.
Then
8i
maps onto the involution
ai · · · a1€
in
as.
In fact, given
another finite set
S'
of finite primes of
JR.(x)
that contains
S,
the same argument
shows that
8i
restricts to an involution in
as'.
As
g
= I~
as',
we get that
8i
S'
is an involution in
9.
o
5. The regular real embedding problem over a PRC field
Let
K
be a PRC field of characteristic
0,
and let
v 0
be a rational number.
We proceed as with PAC fields
[FV2,
Section
1]
with some extra care.
Let
Xt, ... ,
Xm be a partition of X(K) into disjoint clopen subsets. Fix
a closed system X of representatives of the conjugacy classes of involutions in
G(K); then
{K(t)l
t
E
X}
is a closed subset of real closures of K, one for each
ordering of
K
(see Section
1).
Put
Xi= Xnlxi(K/K).
Then
X1
, ...
,Xm
is a
partition of X into disjoint clopen subsets.
LEMMA 5.1. Let A:
11.
--+
it be an unramified Galois cover of absolutely ir-
reducible, non-singular varieties defined over K. Assume that all the automor-
phisms of 'H./it
are
defined over K. Let
(3:
G(K)
--+
Aut('H.jit) be
a
homomor-
phism, and let L be the fixed field ofker(f3). Assume that Lis not formally real.
Let q1, ... , qm
E
11. (
K) satisfy the following.
(1)
t~
=
f3(t)(~),
for each t
E
Xj, for j
=
1, ... , m.
Then there exists
p E
'H.(K) such that
(2) ap
= f3(a)(p) for each a
E
G(K);
(3)
liP-
~II?
:::;
v 2 in K(t), for each t
E
Xj, for j =
1, ... , m;
(4) the point
A(p)
ofit is K-rational and
K(p)
= L.
PROOF. First notice that (4) follows from (2). Indeed, an automorphism of
the unramified cover
11.
--+
it has no fixed points. If (2) holds, then for each
a
E
G(K) we have
a(A(p))
=
A(a(p))::;: A(f3(a)(p))
=
A(p)
and
a(p)
=
p
{:::=:
f3(a)(p)
=
p
{:::=:
f3(a)
=
1
{:::=:
a E G(L).
The rest is a straightforward modification of the proof of
[FV2,
Lemma 1]. Apply
Weil's descent
[W,
Theorem 3] to the maps fr,p = (3(r) o f3(p)-
1
to get a variety
11.'
defined over
K,
and a linear isomorphism
f:
11.'
--+
11.
defined over
L
with these
properties. The map A o
f:
11.'
--+
it is defined over
K
and
a
f
= (3(
a)
o
f,
for
each a E G(K). In particular, suppose that q' E 1-l'(K) and q = f(q') E 1-l(K).
Then, for every
a
E G(K)
(3(a)(q)
=
(f3(a)
o
f)(q')
=
(af)(q')
=
a(f(a-
1
q')).
Conclude that
(5)
aq = f3(a)(q)
{:::=:
q' = aq'
{:::=:
q'
E
1-l'(K(a)).
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