Fields of algebraic numbers with bounded local degrees and their properties

Author:
Sara Checcoli

Journal:
Trans. Amer. Math. Soc. **365** (2013), 2223-2240

MSC (2010):
Primary 11S15, 11R32

DOI:
https://doi.org/10.1090/S0002-9947-2012-05712-6

Published electronically:
September 19, 2012

MathSciNet review:
3009657

Full-text 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 group-theoretical 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****2.**Sara Checcoli and Umberto Zannier,*On fields of algebraic numbers with bounded local degrees*, C. R. Math. Acad. Sci. Paris**349**(2011), no. 1-2, 11–14 (English, with English and French summaries). MR**2755687**, https://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****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****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, Springer-Verlag, Berlin, 1986. MR**868860****6.**Ivo M. Michailov,*Four non-abelian groups of order 𝑝⁴ as Galois groups*, J. Algebra**307**(2007), no. 1, 287–299. MR**2278055**, https://doi.org/10.1016/j.jalgebra.2006.05.021**7.**Władysław Narkiewicz,*Elementary and analytic theory of algebraic numbers*, 2nd ed., Springer-Verlag, Berlin; PWN—Polish Scientific Publishers, Warsaw, 1990. MR**1055830****8.**Jean-Pierre 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****9.**I. R. Shafarevich,*On p-extensions*. Amer. Math. Soc. Transl., Ser.2, 4, 1956, pp. 59-72.**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****11.**Michael Vaughan-Lee,*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****12.**Martin Widmer,*On certain infinite extensions of the rationals with Northcott property*, Monatsh. Math.**162**(2011), no. 3, 341–353. MR**2775852**, https://doi.org/10.1007/s00605-009-0162-7

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, CH-4051 Basel, Switzerland

DOI:
https://doi.org/10.1090/S0002-9947-2012-05712-6

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.