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Brauer groups, embedding problems, and nilpotent groups as Galois groups

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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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Correspondence to Jack Sonn.

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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

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