Wild ramification bounds and simple group Galois extensions ramified only at

Author:
John W. Jones

Journal:
Proc. Amer. Math. Soc. **139** (2011), 807-821

MSC (2010):
Primary 11R21, 11S15

DOI:
https://doi.org/10.1090/S0002-9939-2010-10628-7

Published electronically:
August 12, 2010

MathSciNet review:
2745634

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Abstract: We consider finite Galois extensions of and deduce bounds on the discriminant of such an extension based on the structure of its Galois group. We then apply these bounds to show that there are no Galois extensions of , unramified outside of , whose Galois group is one of various finite simple groups. The set of excluded finite simple groups includes several infinite families.

**[BD08]**Sharon Brueggeman and Darrin Doud,*Local corrections of discriminant bounds and small degree extensions of quadratic base fields*, Int. J. Number Theory**4**(2008), no. 3, 349-361. MR**2424327****[Bru01]**Sharon Brueggeman,*Septic number fields which are ramified only at one small prime*, J. Symbolic Comput.**31**(2001), no. 5, 549-555. MR**1828702 (2002e:11145)****[Dem09]**Lassina Dembélé,*A non-solvable extension of ramified at only*, C. R. Math. Acad. Sci. Paris**347**(2009), 111-116. MR**2538094 (2010g:11191)****[GAP06]**The GAP Group,*GAP - Groups, Algorithms, and Programming, Version 4.4*, 2006,`http://www.gap-system.org`

.**[Har94]**David Harbater,*Galois groups with prescribed ramification*, Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 35-60. MR**1299733 (96a:12008)****[Jon10]**John W. Jones,*Number fields unramified away from*, J. Number Theory**130**(2010), no. 6, 1282-1291.**[JR99]**John W. Jones and David P. Roberts,*Sextic number fields with discriminant*, Number theory (Ottawa, ON, 1996), CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 141-172. MR**2000b:11142****[JR03]**-,*Septic fields with discriminant*, Math. Comp.**72**(2003), no. 244, 1975-1985 (electronic). MR**1986816 (2004e:11119)****[JR06]**-,*A database of local fields*, J. Symbolic Comput.**41**(2006), no. 1, 80-97. MR**2194887 (2006k:11230)****[Les]**Sylla Lesseni,*The nonexistence of nonsolvable octic number fields ramified only at one small prime*, Math. Comp.**75**(2006), 1519-1526. MR**2219042 (2007d:11121)****[Mal04]**Sergey Malyushitsky,*On Sylow -subgroups of finite simple groups of order up to*, Ph.D. thesis, The Ohio State University, 2004.**[Mar63]**G. N. Markšaĭtis,*On -extensions with one critical number*, Izv. Akad. Nauk SSSR Ser. Mat.**27**(1963), 463-466. MR**0151452 (27:1437)****[Moo00]**Hyunsuk Moon,*Finiteness results on certain mod Galois representations*, J. Number Theory**84**(2000), no. 1, 156-165. MR**1782427 (2001g:11082b)****[Moo07]**-,*On four-dimensional mod Galois representations and a conjecture of Ash et al*. Bull. Korean Math. Soc.**44**(2007), no. 1, 173-176. MR**2297707 (2007m:11157)****[Odl76]**Andrew Odlyzko,*Table 2: Unconditional bounds for discriminants*,`http://www.dtc.umn.edu/~odlyzko/unpublished/discr.bound.table2`, 1976.**[Odl90]**A. M. Odlyzko,*Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results*, Sém. Théor. Nombres Bordeaux (2)**2**(1990), no. 1, 119-141. MR**1061762 (91i:11154)****[Ser79]**Jean-Pierre Serre,*Local fields*, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979, translated from the French by Marvin Jay Greenberg. MR**82e:12016****[Ser86]**-,*Œuvres. Vol. III*, Springer-Verlag, Berlin, 1986, 1972-1984. MR**926691 (89h:01109c)****[Tat94]**John Tate,*The non-existence of certain Galois extensions of unramified outside*, Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 153-156. MR**95i:11132**

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Additional Information

**John W. Jones**

Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, Arizona 85287

Email:
jj@asu.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10628-7

Received by editor(s):
April 2, 2010

Published electronically:
August 12, 2010

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.