28
MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
M.
Let
N
be the algebraic closure of
Min F.
Let
H:::;
G(F/E)
such that
HE ImG(M) and resNH
=
G(N/M).
Then there exists an M-homomorphism
'1/J: S--+ M
such that
'1/;(R)
=
M
and
'1/J*(G(M)) =H.
PROPOSITION 8.2.
Let M be
a
PRC field.
If
G(M) has the embedding
property, then M is real Frobenius.
PROOF. (Cf.
[HL,
Proposition 5.6].) Let
8/R, F/E, N,
and
H
be
as
in
Definition 8.1. The embedding property gives an epimorphism of involutory
structures
h: G(M)--+
H with resN o
h
=
resN. Put
L
=
MF.
Then
G(L/E)
=
G(ME/E)
xa(NE/E)
G(F/E)
=
G(M)
xa(N/M)
G(F/E).
Let
D
be the fixed field
of~
=
{(8, h(8))1 8
E
G(M)}
in
L.
Then
D/M
is
regular,
DF
=
DM
=
L,
and
D
n
F
=
E [FJ,
p. 354]. We show that
D/M
is
totally real. Let
P
be an ordering on
M.
There is
E
E
I(M)
such that Pis the
restriction of
P€
from
M(t:).
Then
h(t:)
E
IH
~
I(F/E).
Observe that
M(t:)
and
F(h(t:))
are linearly disjoint over
N(t:)
and
L(t:, h(t:))
=
D(t:)F(h(t:))
contains
D.
By assumption there is an ordering
Q
of
F(h(t:))
such that resN(E)Q
=
resN(E)P€.
Therefore
P€
and
Q
extend to an ordering of
L(t:, h(t:)) [J,
p. 241]. The restriction
of this ordering to
D
extends
P.
The integral closure U of R in D is finitely generated over M
[F J,
p. 354] and
hence
U
is the coordinate ring of an absolutely irreducible variety
V
defined over
M.
Since~
is PRC,
the~
exists _!l:n M-homomorphism '¢0
:
U
--+
M.
Extend
'¢0
to an M-epimorphism
MU--+ M,
and let '¢:
S--+ M
be its restriction to
S.
Then
'1/;(R)
=
M,
and, by
[FHJ,
Remark on p. 9], we may arrange it so that
'1/J*:G(M)--+ G(F/E)
coincides with
h.
Therefore
'1/J*(G(M)) =H.
o
By Corollary 7.7 and Lemma 7.3(c), G(Qtr) has the embedding property. By
[P], Qtr is PRC. Therefore:
COROLLARY 8.3. Qtr
is real Frobenius.
9. Real Galois Stratification
This section gives a quantifier elimination procedure f.::r the theory of real
Frobenius fields in the language below. The procedure is similar to
[FJ,
Chapter
25] and almost the same
as
in
[HL].
So we only comment on the differences.
A Galois ring/set cover
C/A
over a field
K [FJ,
p. 403] is real if
A
is
nonsingular, char(K)
=
0, and
K(C)
is not formally real. Put
G(C/A)
=
G(K(C)/K(A))
(Example 7.1(a)) and let Sub[C/A] be the involutory substruc-
tures of
G(C/A).
Let
K
~
M
be a field. Each a E
A(M)
determines a K-homomorphism
¢: K[A]
--+
M,
and therefore (see Section 6) a homomorphism ¢*:
G(M)
--+
G(C/A)
(unique up to an inner automorphism of
G(C/A)).
Example 7.1(b)
says that
¢*(G(M) :::; G(C/A).
Omitting the reference to
C
and
M,
define
the Artin symbol
Ar(A,a)
as the set {¢*(G(M)""I u E
G(C/A)}.
This is a
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