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)
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