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