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Number fields with solvable Galois groups and small Galois root discriminants

Authors: John W. Jones and Rachel Wallington
Journal: Math. Comp. 81 (2012), 555-567
MSC (2010): Primary 11R21; Secondary 11R37
Published electronically: June 3, 2011
MathSciNet review: 2833508
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Abstract | References | Similar Articles | Additional Information

Abstract: We apply class field theory to compute complete tables of number fields with Galois root discriminant less than $ 8\pi e^{\gamma}$. This includes all solvable Galois groups which appear in degree less than $ 10$, groups of order less than $ 24$, and all dihedral groups $ D_p$ where $ p$ is prime.

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  • [BM83] Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863-911. MR 84f:20005
  • [CHM98] John H. Conway, Alexander Hulpke, and John McKay, On transitive permutation groups, LMS J. Comput. Math. 1 (1998), 1-8 (electronic). MR 99g:20011
  • [Coh00] Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 2000k:11144
  • [FK03] Claus Fieker and Jürgen Klüners, Minimal discriminants for fields with small Frobenius groups as Galois groups, J. Number Theory 99 (2003), no. 2, 318-337. MR 1968456 (2004f:11147)
  • [GAP06] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4, 2006, (
  • [Jon] John W. Jones, Wild ramification bounds and simple group Galois extensions ramified only at $ 2$, Proc. Amer. Math. Soc. 139 (2011), no. 3, 807-821. MR 2745634
  • [JR03] John W. Jones and David P. Roberts, Septic fields with discriminant $ \pm2\sp a3\sp b$, Math. Comp. 72 (2003), no. 244, 1975-1985 (electronic). MR 1986816 (2004e:11119)
  • [JR07a] -, Galois number fields with small root discriminant, J. Number Theory 122 (2007), no. 2, 379-407. MR 2292261 (2008e:11140)
  • [JR07b] -, Website: Number fields with small grd,, 2007.
  • [PAR08] The PARI Group, Bordeaux, Pari/gp, version 2.3.4, 2008.
  • [Per77] Robert Perlis, On the equation $ \zeta _{K}(s)=\zeta _{K'}(s)$, J. Number Theory 9 (1977), no. 3, 342-360. MR 0447188 (56:5503)
  • [Sch89] Leila Schneps, $ \tilde{D}_4$ et $ \hat D_4$ comme groupes de Galois, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 2, 33-36. MR 980080 (90f:12004)
  • [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)

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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

Rachel Wallington
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, Arizona 85287
Address at time of publication: Faith Christian School, P.O. Box 31300, Mesa, Arizona 85275

Received by editor(s): July 24, 2010
Received by editor(s) in revised form: December 15, 2010
Published electronically: June 3, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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