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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $K$.
- Emil Artin, Algebraic numbers and algebraic functions, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR 0237460
- Lior Bary-Soroker, Dan Haran, and David Harbater, Permanence criteria for semi-free profinite groups, Math. Ann. 348 (2010), no. 3, 539–563. MR 2677893, DOI https://doi.org/10.1007/s00208-010-0484-8
- Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
- J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410
- Pierre Dèbes and Bruno Deschamps, The regular inverse Galois problem over large fields, Geometric Galois actions, 2, London Math. Soc. Lecture Note Ser., vol. 243, Cambridge Univ. Press, Cambridge, 1997, pp. 119–138. MR 1653011, DOI https://doi.org/10.1017/CBO9780511666124.007
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
- Michael D. Fried and Moshe Jarden, Field arithmetic, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008. Revised by Jarden. MR 2445111
- Arno Fehm and Elad Paran, Galois theory over rings of arithmetic power series, Adv. Math. 226 (2011), no. 5, 4183–4197. MR 2770445, DOI https://doi.org/10.1016/j.aim.2010.11.010
- Arno Fehm and Elad Paran, Klein approximation and Hilbertian fields, J. Reine Angew. Math. 676 (2013), 213–225. MR 3028759, DOI https://doi.org/10.1515/crelle.2012.007
- David Harbater, Algebraic rings of arithmetic power series, J. Algebra 91 (1984), no. 2, 294–319. MR 769575, DOI https://doi.org/10.1016/0021-8693%2884%2990104-2
- David Harbater, Convergent arithmetic power series, Amer. J. Math. 106 (1984), no. 4, 801–846. MR 749258, DOI https://doi.org/10.2307/2374325
- David Harbater, Galois covers of an arithmetic surface, Amer. J. Math. 110 (1988), no. 5, 849–885. MR 961498, DOI https://doi.org/10.2307/2374696
- Dan Haran and Moshe Jarden, Regular split embedding problems over complete valued fields, Forum Math. 10 (1998), no. 3, 329–351. MR 1619723, DOI https://doi.org/10.1515/form.10.3.329
- Dan Haran and Moshe Jarden, Regular split embedding problems over function fields of one variable over ample fields, J. Algebra 208 (1998), no. 1, 147–164. MR 1643991, DOI https://doi.org/10.1006/jabr.1998.7454
- David Harbater and Katherine F. Stevenson, Local Galois theory in dimension two, Adv. Math. 198 (2005), no. 2, 623–653. MR 2183390, DOI https://doi.org/10.1016/j.aim.2005.06.011
- Dan Haran and Helmut Völklein, Galois groups over complete valued fields, Israel J. Math. 93 (1996), 9–27. MR 1380632, DOI https://doi.org/10.1007/BF02761092
- Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859
- Elad Paran, Algebraic patching over complete domains, Israel J. Math. 166 (2008), 185–219. MR 2430432, DOI https://doi.org/10.1007/s11856-008-1027-9
- Elad Paran, Split embedding problems over complete domains, Ann. of Math. (2) 170 (2009), no. 2, 899–914. MR 2552112, DOI https://doi.org/10.4007/annals.2009.170.899
- Jérôme Poineau, Raccord sur les espaces de Berkovich, Algebra Number Theory 4 (2010), no. 3, 297–334 (French, with English and French summaries). MR 2602668, DOI https://doi.org/10.2140/ant.2010.4.297
- Florian Pop, Embedding problems over large fields, Ann. of Math. (2) 144 (1996), no. 1, 1–34. MR 1405941, DOI https://doi.org/10.2307/2118581
- Florian Pop, Henselian implies large, Ann. of Math. (2) 172 (2010), no. 3, 2183–2195. MR 2726108, DOI https://doi.org/10.4007/annals.2010.172.2183
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Additional Information
Arno Fehm
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
MR Author ID:
887431
ORCID:
0000-0002-2170-9110
Elad Paran
Affiliation:
Department of Mathematics and Computer Science, Open University of Israel, 43107 Raanana, Israel
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