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)).