Fields of algebraic numbers with bounded local degrees and their properties
Author:
Sara Checcoli
Journal:
Trans. Amer. Math. Soc. 365 (2013), 22232240
MSC (2010):
Primary 11S15, 11R32
Published electronically:
September 19, 2012
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results in Checcoli and Zannier's paper (2011) and obtaining relevant generalizations for them. In particular we show that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at all but finitely many primes; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of an infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of , which is defined as the compositum of all number fields of degree at most over ; being generated by elements of uniformly bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on grouptheoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.
 1.
Enrico
Bombieri and Umberto
Zannier, A note on heights in certain infinite extensions of
ℚ, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend.
Lincei (9) Mat. Appl. 12 (2001), 5–14 (2002)
(English, with English and Italian summaries). MR 1898444
(2003d:11155)
 2.
Sara
Checcoli and Umberto
Zannier, On fields of algebraic numbers with bounded local
degrees, C. R. Math. Acad. Sci. Paris 349 (2011),
no. 12, 11–14 (English, with English and French summaries). MR 2755687
(2012a:11167), http://dx.doi.org/10.1016/j.crma.2010.12.007
 3.
Klaus
Doerk and Trevor
Hawkes, Finite soluble groups, de Gruyter Expositions in
Mathematics, vol. 4, Walter de Gruyter & Co., Berlin, 1992. MR 1169099
(93k:20033)
 4.
I.
B. Fesenko and S.
V. Vostokov, Local fields and their extensions, 2nd ed.,
Translations of Mathematical Monographs, vol. 121, American
Mathematical Society, Providence, RI, 2002. With a foreword by I. R.
Shafarevich. MR
1915966 (2003c:11150)
 5.
Michael
D. Fried and Moshe
Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer
Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
vol. 11, SpringerVerlag, Berlin, 1986. MR 868860
(89b:12010)
 6.
Ivo
M. Michailov, Four nonabelian groups of order 𝑝⁴ as
Galois groups, J. Algebra 307 (2007), no. 1,
287–299. MR 2278055
(2008a:12006), http://dx.doi.org/10.1016/j.jalgebra.2006.05.021
 7.
Władysław
Narkiewicz, Elementary and analytic theory of algebraic
numbers, 2nd ed., SpringerVerlag, Berlin, 1990. MR 1055830
(91h:11107)
 8.
JeanPierre
Serre, Topics in Galois theory, Research Notes in Mathematics,
vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes
prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the
author. MR
1162313 (94d:12006)
 9.
I. R. Shafarevich, On pextensions. Amer. Math. Soc. Transl., Ser.2, 4, 1956, pp. 5972.
 10.
I.
R. Šafarevič, Construction of fields of algebraic
numbers with given solvable Galois group, Izv. Akad. Nauk SSSR. Ser.
Mat. 18 (1954), 525–578 (Russian). MR 0071469
(17,131d)
 11.
Michael
VaughanLee, The restricted Burnside problem, 2nd ed., London
Mathematical Society Monographs. New Series, vol. 8, The Clarendon
Press Oxford University Press, New York, 1993. MR 1364414
(98b:20047)
 12.
Martin
Widmer, On certain infinite extensions of the rationals with
Northcott property, Monatsh. Math. 162 (2011),
no. 3, 341–353. MR 2775852
(2012f:11123), http://dx.doi.org/10.1007/s0060500901627
 1.
 E. Bombieri, U. Zannier, A note on heights in certain infinite extensions of . Rend. Mat. Acc. Lincei, 12, 2001, pp. 514. MR 1898444 (2003d:11155)
 2.
 S. Checcoli, U. Zannier, On fields of algebraic numbers with bounded local degrees. C. R. Math. Acad. Sci. Paris 349, 2011, pp. 1114. MR 2755687 (2012a:11167)
 3.
 K. Doerk, T. Hawkes, Finite Solvable Groups. De Gruyter, Berlin, 1992. MR 1169099 (93k:20033)
 4.
 I. Fesenko, S. Vostokov, Local fields and their extensions, Amer. Math. Soc., Providence, R.I., second edition, 2002. MR 1915966 (2003c:11150)
 5.
 M. D. Fried, M. Jarden, Field Arithmetic, SpringerVerlag, Berlin, 1986. MR 868860 (89b:12010)
 6.
 I. Michailov, Four nonabelian groups of order as Galois groups. J. Algebra 307, 2007, pp. 287299. MR 2278055 (2008a:12006)
 7.
 W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, SpringerVerlag, Berlin, 1990. MR 1055830 (91h:11107)
 8.
 JP. Serre, H. Darmon, Topics in Galois Theory. Research Notes in Mathematics, Jones and Bartlett Publishers, 1992. MR 1162313 (94d:12006)
 9.
 I. R. Shafarevich, On pextensions. Amer. Math. Soc. Transl., Ser.2, 4, 1956, pp. 5972.
 10.
 I. R. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group. Izv. Akad. Nauk SSSR Ser. Mat., 18:6, 1954, pp. 525578. MR 0071469 (17:131d)
 11.
 M. R. VaughanLee, The restricted Burnside problem. Second Ed., Oxford University Press, 1993. MR 1364414 (98b:20047)
 12.
 M. Widmer, On certain infinite extensions of the rationals with Northcott property. Monatsh. Math., 2009. MR 2775852
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
11S15,
11R32
Retrieve articles in all journals
with MSC (2010):
11S15,
11R32
Additional Information
Sara Checcoli
Affiliation:
Mathematisches Institut, University of Basel, Rheinsprung 21, CH4051 Basel, Switzerland
DOI:
http://dx.doi.org/10.1090/S000299472012057126
PII:
S 00029947(2012)057126
Received by editor(s):
December 21, 2010
Received by editor(s) in revised form:
March 14, 2011, March 29, 2011, April 7, 2011, July 15, 2011, August 31, 2011, and September 18, 2011
Published electronically:
September 19, 2012
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
