24 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN

there exists a

K-

homomorphism ¢:

R _.... K

with ¢* (

G (

K))

=

G (FIE)

and

¢*(lxj(K))

=

!Qj(FIE),

for each 1

~

j

~

m.

Thus, ¢*(I(K))

=!(FIE).

COROLLARY 6.5. If

K is a number field, then K is totally real Hilbertian.

PROOF. Consider (2). For each j put

Pj

=

resKQj.

Since IX(K)I oo, we

may refine

X

1

, ... ,

Xm

to assume

Xj

= {

Pj}.

As K is dense in each of its real

closures, Lemma 6.3 gives a nonempty open subset

Uj

of K (with respect to

Pj)

so that

¢~(I

Pj (

K)) ;:;

I Q J (FIE)

for each

a

E

Uj.

Consider the Hilbert set

HK ={a

E

Kl

h(a)-::/=

0 and ¢:(G(K))

=

G(FIE)}

[FJ,

Lemma 12.12]. By

[G,

Lemma 3.4],

HK

is dense in K in the product

topology induced by

p1' ... 'Pn,

that is, there is

a

E

HK

n

u1

n ... n

Um.

Observe that

lQ1 (FIE)

is a conjugacy class in

G(F I E).

The surjectivity

of¢~

implies that

¢~(Ip 1 (K))

=

IQJ(FIE).

o

PROPOSITION 6.6.

Let K

=

Qtr.

Then K is totally real Hilbertian.

PROOF. Assume

(2).

Let

z

be a primitive element for the cover

SIR.

Let

K'

;:;

K

be a number field such that

h(x)

E

K'[x]. Put

R'

=

K'[x,

h(x)-

1]

and

S'

=

R'[z],

and let

E'

and

F'

be their quotient fields. For each j let

Qj

be

the restriction of

Qj

to

E',

and let

Xj

be the restriction of

Xj

to

K'.

Take

K'

sufficiently large to assume the following.

(i)

S'

I

R'

is a real Galois cover.

(ii)

[F' : E']

=

[F: E],

and therefore

K

and

F'

are linearly disjoint over

K'.

(iii) The sets

Xi, ... , x:r,

are distinct.

Then Xi, ...

,x:r,

is a partition of

X(K')

and

G(F'IE')

~

G(FIE).

By Corollary 6.4 there exists a E

K'

such that an extension

'¢': S' _.... K

of

¢a:R' _.... K'

satisfies 'lj;'*(Ix'(K))

=

IQ'(F'IE').

Extend

cPa

to the

K-

J J

homomorphism ¢a:

R _.... K.

As

K

and

S'

are linearly disjoint over

K',

it is

possible to extend this

cPa

and'¢' to the same K-homomorphism

'¢: S _....

K.

By

(1), the following diagram commutes.

G(FI

E)

---:;v-

G(K)---

I(K)

!resp' ! !

G(F' IE')

~

G(K') ---

I(K')

From (ii), the left vertical map is an isomorphism.

AsK=

Qtr,

I(K)

=

I(Q)

=

I(K').

Thus, the right vertical inclusion is surjective and maps

lx1

(K) onto

lx;(K').

Diagram chasing yields

'lj;*(IxJ(K))

=

IQJ(FIE).

But

G(FIE)

=

(U;'=

1

IQJFI E)),

and hence 'lf;*(G(K))

=

G(FI E).

o

7. Absolute Galois group of the totally real algebraic numbers

In this section we consider the following category. An involutory structure

is a pair

(G, 10

)

=

G for short, where

G

is a profinite group and

Ic

is a closed