REAL HILBERTIANITY-TOTALLY REAL NUMBERS

29

conjugacy class in Sub[C/A]. For properties of the Artin symbol see [FJ, Section

5.3].

For n

~

0 let 1r: An+l

---tAn

be the projection on the first n coordinates. Let

A

~

A

n+l

and

B

~

An

be two non-singular basic sets

[F

J,

p. 244] such that

1r(A) =B. Then K[B]

~

K[A]. Let x and (x,

y)

be generic points of Band A,

respectively. Then K(A) = K(B)(y). Furthermore, let C/A and D/B be real

Galois covers such that K(D) contains the algebraic closure of K(B) in K(C).

DEFINITION 9.1 [HL, Definition 7.1]. Let M be a field extension of K. An

M-specialization of the pair (C/A, D/B) is a K-homomorphism ¢from C into

an overfield of M with these properties: ¢(K[B])

~

M; and if y is transcendental

over K(B), then ¢(y) is transcendental over

M.

For such a specialization put y' = ¢(y), N = M[¢{D)], R = M[¢(K[A])],

E = M(y') {the quotient field of R), S = M[¢(C)], and F = E[¢(C)] {the

quotient field of S). Then¢ induces an embedding ¢*:G(F/E)

---t

G(C/A).

Assume that dim A = dim B + 1. The pair ( C /A, D /B) is specializa-

tion compatible if the following properties hold for every M and each M-

specialization ¢ as above.

{i) K(D) is the algebraic closure of K(B) in K{C). and for every M and

each M-specialization ¢as above [K{C) : K(D)(y)] = [F:

N(y')].

(ii) The cover K(C)/K(A) is amply real over K(B).

{iii) For each involution t E G(F/E) with ¢*{t) real the extension F(t)/N(t)

is totally real.

Assume that dim A = dim B. The pair ( C /A, D /B) is said to be specializa-

tion compatible if K[A] is integral over K[B] and C =D.

LEMMA 9.2. Assume that dimA =dimE.+ 1 and that (C/A,D/B) is spe-

cialization compatible. Let Con(A) be a conjugacy domain in Sub[C/A], and let

S

be a a set of (isomorphism types of) involutory structures. Define

Con{B) =

{

resK(D)(SnCon{A)) ifdimA=dimB+1;

{Gal G E Con(A), a E G{C/B)} ifdimA =dimE.

Let M be a real F'robenius field that contains K, and let

b

E

B(M). Assume

lmG{M) n Sub[C/A] = S. Then Ar(B, b)

~

Con(B) if and only if there is

a E A(M) such that 1r(a) =

b

and Ar(A, a)

~

Con{A).

PROOF. See [HL, Lemma 7.2] in case dim A= dimE+ 1 and [HL, Lemma

7.3] in case dimA =dimE. (Replace everywhere thee-structures of [HL] by

our involutory structures.) o

LEMMA 9.3 [HL, Lemma 7.5]. Let K

1

be a finite extension of K(D). There

are Zariski open subsets A'

~

A, B'

~

B and a specialization compatible pair of

real Galois covers (C' /A' , D' / B') such that K(C)

~

K(C') and K

1

~

K(D').

From now on we can proceed exactly as in

[F

J,

Chapter 25]. Replace Galois

covers with real Galois covers, and conjugacy classes of subgroups of G(Ci/Ai)