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
is decidable. Since the ring of integers of
is undecidable, this gives a
natural undecidable ring whose quotient field is decidable.
The theory and use in [F] of moduli spaces of covers of the Riemann sphere
with prescribed ramification data has been further developed in
the main theme is that K-rational points of the moduli spaces correspond to
covers defined over
notes a correspondence between
existence of K-rational points on certain related spaces and the solvability of
regular embedding problems over
Thus, using moduli spaces allows us to
prove solvability of regular embedding problems over fields
suitably large for
such varieties to have the requisite K-.rational points.
This principle appears in
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
Recall [FJ, p. 129] that
PAC (pseudo algebraically closed)
absolutely irreducible variety V defined over K has a K-rational point. Fur-
PRC (pseudo real closed)
if every absolutely irreducible
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
$.25 per page