Abstract
Let ℚ ab denote the maximal abelian extension of the rationals ℚ, and let ℚabnil denote the maximal nilpotent extension of ℚ ab . We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚabnil(t). We also prove that ℚabnil is not PAC (pseudo-algebraically closed).
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References
[AG] M. Auslander and O. Goldman,The Brauer group of a commutative ring, Trans. Amer. Math. Soc.97 (1960), 367–409.
[FM] M. Fried and M. Jarden,Field Arithmetic, Springer-Verlag, Berlin, 1986.
[FS] B. Fein and M. Schacher,Brauer groups of rational function fields, in Groupe de Brauer, Lecture Notes in Math.844, Springer-Verlag, Berlin, 1981.
[FV] M. Fried and H. Völklein,The inverse Galois problem and rational points on modular spaces, Math. Ann.290 (1991), 771–800.
[H] K. Hoechsmann,Zum Einbettungsproblem, J. Reine Angew. Math.229 (1968), 81–106.
[J] M. Jarden,Intersections of local algebraic extensions of a Hilbertian field, inGenerators and Relations in Groups and Geometries (A. Barlotti et al., eds.), Kluwer, Dordrecht, 1991.
[M] B.H. Matzat,Konstruktive Galoistheorie, Lecture Notes in Mathematics1284, Springer-Verlag, Berlin, 1987.
[N] J. Neukirch,Über das Einbettungsproblem der algebraische Zahlentheorie, Inv. Math.21 (1973), 59–116.
[Ri] L. Ribes,Introduction of profinite groups and Galois cohomology, Queens Papers in Pure and Applied Math, 1970.
[Rib] P. Ribenboim,Theorie des Valuations, Sem. Math. Sup., University of Montreal Press, 1968.
[Se] J.P. Serre,Local Fields, Springer-Verlag, Berlin, 1979.
[Se1] J.P. Serre,Topics in Galois Theory, Lecture notes, Harvard University, 1988.
[Sh] I.R. Shafarevich, On construction of fields with a given Galois group of order ℓα, Transl. Amer. Math. Soc., Ser. 2,4 (1956), 107–142.
[So] J. Sonn,On Brauer groups and embedding problems over rational function fields, J. Algebra131 (1990), 631–640.
[W] E. Weiss,Algebraic Number Theory, McGraw-Hill, New York, 1963.
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This paper was inspired by the author's participation in a special year on the arithmetic of fields at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I would like to express my appreciation to the Institute for its hospitality, and to the organizers, especially Moshe Jarden.
Partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-Japan Technion Society Research Fund.
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Sonn, J. Brauer groups, embedding problems, and nilpotent groups as Galois groups. Israel J. Math. 85, 391–405 (1994). https://doi.org/10.1007/BF02758649
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DOI: https://doi.org/10.1007/BF02758649