Sliceable groups and towers of fields
-
Sigrid Böge
,
Moshe Jarden
Abstract
Let l be a prime number, K a finite extension of ℚl,
and D a finite-dimensional central division algebra over
K.
We prove that the profinite group
Funding source: Minerva Foundation
Award Identifier / Grant number: Minkowski Center for Geometry at Tel Aviv University
Funding statement: The second author has been supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation, and by an ISF grant. The third author acknowledges “Dr. Max Rössler, the Walter Haefner Foundation”, the ETH Foundation, and the ETH Institute for Theoretical Studies for support and hospitality. In addition, the author acknowledges support by ISF and NSF.
We would like to thank Wulf-Dieter Geyer, Michael Larsen, Andrei Rapinchuk, Aharon Razon, and David Zywina for helpful communications and advice.
References
1 E. Artin, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York, 1967. Search in Google Scholar
2 A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991. 10.1007/978-1-4612-0941-6Search in Google Scholar
3 N. Bourbaki, Lie Groups and Lie Algebras, Chapter 1–3, Springer, Berlin, 1989. 10.1515/9780691223803-002Search in Google Scholar
4 N. Bourbaki, Algebra II, Chapters 4–7, Springer, Berlin, 1990. Search in Google Scholar
5 J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, London, 1967. Search in Google Scholar
6 J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-p Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge University Press, Cambridge, 1999. 10.1017/CBO9780511470882Search in Google Scholar
7 B. Farb and R. K. Dennis, Noncommutative Algebra, Grad. Texts in Math. 144, Springer, New York, 1993. 10.1007/978-1-4612-0889-1Search in Google Scholar
8 M. D. Fried and M. Jarden, Field Arithmetic, 3rd ed., Ergeb. Math. Grenzgeb. (3) 11, Springer, Heidelberg, 2008. Search in Google Scholar
9 G. J. Janusz, Algebraic Number Fields, Academic Press, New York, 1973. Search in Google Scholar
10 W. Jehne, Kronecker classes of algebraic number fields, J. Number Theory 9 (1977), 279–320. 10.1016/0022-314X(77)90066-XSearch in Google Scholar
11 W. M. Kantor, A. Lubotzky and A. Shalev, Invariable generation of infinite groups, J. Algebra 421 (2015), 296–310. 10.1016/j.jalgebra.2014.08.030Search in Google Scholar
12 H. Koch, Galoissche Theorie der p-Erweiterungen, Math. Monogr. 10, VEB Deutscher Verlag der Wissenschaften, Berlin, 1970. 10.1007/978-3-642-92997-7Search in Google Scholar
13 S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, 1970. Search in Google Scholar
14 S. Lang, Elliptic Functions, Addison-Wesley, Reading, 1973. Search in Google Scholar
15 S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, 1993. Search in Google Scholar
16 M. Larsen and R. Ramakrishna, The inverse Galois problem for p-adic Lie algebras, work in progress. Search in Google Scholar
17 L. Ribes and P. Zalesskii, Profinite Groups, Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2000. 10.1007/978-3-662-04097-3Search in Google Scholar
18 J.-P. Serre, Propriétés galoisiennes des points d'ordere fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331. 10.1007/BF01405086Search in Google Scholar
19 J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Math. 1500, Springer, Berlin, 1992. 10.1007/978-3-540-70634-2Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
Articles in the same Issue
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
