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Split embedding problems over the open arithmetic disc


Authors: Arno Fehm and Elad Paran
Journal: Trans. Amer. Math. Soc. 366 (2014), 3535-3551
MSC (2010): Primary 12E30, 12F12, 13J05
DOI: https://doi.org/10.1090/S0002-9947-2014-05931-X
Published electronically: February 26, 2014
MathSciNet review: 3192606
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Abstract: Let $ \mathbb{Z}\{t\}$ be the ring of arithmetic power series that converge on the complex open unit disc. A classical result of Harbater asserts that every finite group occurs as a Galois group over the quotient field of $ \mathbb{Z}\{t\}$. We strengthen this by showing that every finite split embedding problem over $ \mathbb{Q}$ acquires a solution over this field. More generally, we solve all $ t$-unramified finite split embedding problems over the quotient field of $ \mathcal {O}_K\{t\}$, where $ \mathcal {O}_K$ is the ring of integers of an arbitrary number field $ K$.


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Additional Information

Arno Fehm
Affiliation: Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

Elad Paran
Affiliation: Department of Mathematics and Computer Science, Open University of Israel, 43107 Raanana, Israel

DOI: https://doi.org/10.1090/S0002-9947-2014-05931-X
Received by editor(s): August 29, 2011
Received by editor(s) in revised form: August 2, 2012
Published electronically: February 26, 2014
Additional Notes: This research was supported by the DFG program “Initiation and Intensification of Bilateral Cooperation”
Article copyright: © Copyright 2014 American Mathematical Society