REAL HILBERTIANITY-TOTALLY REAL NUMBERS
15
Finally, assume
G
has a non-inner automorphism f of order 2. Let GI(E} be
the subgroup of Aut(G) generated by
G
and
f.
In particular, the centralizer of
Gin GI(E} is trivial. Let
I~
GI(E}-....G be a set of involutions, with f E
I
and
III;:::
2. Lemma 3.1 (with
m
replaced by 2m) produces an r-tuple (crt, ... ,err) E
£(r)(G) (see (2.12)) with the
fo~lowing
properties:
(3)
f
-1 -1 -1
£
h 1
.
cri = cr1 · · ·
CTj-1CTj
cri-l · · · cr1 , tOr eac _
J _
e;
(4)
cr~+i
= cr;:(2m+l-i) for j = 1, ... , 2m;
(5)
I=
{E,
ECTt,
ECT1cr2, ... , fCT1cr2 ···ere}·
Fix (for each f and each
I)
such an r-tuple (cr1, ... ,err)· As cr1 ··
·err=
1, there
is a unique epimorphism f
0
:
II1
(IP'1-....b,
0)
---+
G
with
fo(''"'"/j)
= cri, for j = 1, ... , r.
DEFINITION 4.1. The point q = [b, 0,
fo]
E
1i
is called the basic point
associated with
G,
E,
and
I.
The neighborhood
N
= {p = [a, 0,
!]
E 'Hinlla
n
Djl
= 1,
f('"'/j)
=
fo(··u)
=
CTj,
for j = 1, ... 'r}
of q in
1i
is called the basic neighborhood of q.
REMARK 4.2.
Properties of
a
basic neighborhood.
(a) A priori,
N
is a neighborhood of q in 'Hin (see
(2.6)).
Yet,
N
is connected.
Hence,
N
~
'H.
(b) The point b is Q-rational
(2.7).
Hence q is algebraic over Q.
(c) Let p E
N,
and let a = { a 1, ... ,
ar}
= W (p). Then without loss of gen-
erality
ai
E
Dj,
for j = 1, ... ,
r.
If a is JR-rational (i.e.,
(X- ai) ···(X-
ar)
E JR[X)), then a1
· · ·
ae
are real, and a&+(2m+l-j) is the complex
conjugate of
ae+j,
for j
=
1, ... , m.
(d) Let c be the complex conjugation. As
1i
is an affine variety, we may
embed it in a fixed affine space
An.
Then the complex topology on it
is given by the norm
II -
lie defined in Definition 1.4. There are only
finitely many choices of
E
and
I.
Hence there are only finitely many basic
points associated with
G.
Thus there is a positive rational number v
(that depends only on G) such that if q is a basic point, p E
'H,
and
liP - qllc
2
v
2
,
then p is in the basic neighborhood
N
of q.
LEMMA 4.3.
Let
p
EN
such that
W(p)
is JR.-rational. Then
o,(p) = c(p),
where
c
is complex conjugation.
PROOF. Write p as [a, 0, f). Then a
=
W(p). We have o,(p)
=
[a, 0,
Eo
f)
by (2.14) and c(p) = [c(a), 0, cf] = [a, 0, cf] by (2.11). It remains to show that
cf =Eo
f.
Observe that
c{Ji
=
f3j
1
,
for j
=
1, ... ,
e.
Recursively:
(6)
-1 -1 -1 -1 -1 -1
C')'1 = 1'1 ' C'Y2 = 1'11'2 1'1 · · 'C'Ye = 1'1 · · · 'Ye-1'Ye 'Ye-1 · · · 1'1 ·
Furthermore,
(7)
C'Ye+j
=
'Y;:(2m+ 1-j)' for j
=
1, ... , 2m.
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