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.