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
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