Contemporary Mathematics
Volume 174, 1994
Real Hilbertianity and the
Field of Totally Real Numbers
MICHAEL D. FRIED*t, DAN HARAN*t
AND HELMUT VOLKLEIN*"
ABSTRACT. We use moduli spaces for covers of the Riemann sphere to solve
regular embedding problems, with prescribed extendability of orderings,
over PRC fields. As a corollary we show that the elementary theory of
Qtr
is decidable. Since the ring of integers of
Qtr
is undecidable, this gives a
natural undecidable ring whose quotient field is decidable.
Introduction
The theory and use in [F] of moduli spaces of covers of the Riemann sphere
with prescribed ramification data has been further developed in
[FVl].
There
the main theme is that K-rational points of the moduli spaces correspond to
covers defined over
K.
Furthermore,
[FV2]
notes a correspondence between
existence of K-rational points on certain related spaces and the solvability of
regular embedding problems over
K.
Thus, using moduli spaces allows us to
prove solvability of regular embedding problems over fields
K
suitably large for
such varieties to have the requisite K-.rational points.
This principle appears in
[FV2]
to show that the absolute Galois group of a
countable Hilbertian PAC field of characteristic 0 is free. The natural extension
of this to the (larger) class of Hilbert ian PRC fields appears in
[FV3].
Recall [FJ, p. 129] that
K
is
PAC (pseudo algebraically closed)
if every
absolutely irreducible variety V defined over K has a K-rational point. Fur-
thermore
[P2], K
is
PRC (pseudo real closed)
if every absolutely irreducible
1991 Ma:hematics
Subject
Classification. Primary 12D15, 12E25, 12F12, 11G25.
* Support from the Institute for Advanced Study at Hebrew University, 1991-92.
t Supported by NSA grant MDA 14776 and BSF grant 87-00038.
+
Supported by Max-Planck-Institut flir Mathematik, Bonn, 1992-93.
Supported by NSA grant MDA 904-89-H-2028.
This paper is in final form and no version of it will be submitted for publication elsewhere.
©
1994 American Mathematical Society
0271-4132/94 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/174/01849
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