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
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