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Exceptional units in a family of quartic number fields
Author(s):
G.
Niklasch;
N.
P.
Smart.
Journal:
Math. Comp.
67
(1998),
759-772.
MSC (1991):
Primary 11D61, 11R27, 11J86, 11J25
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Abstract:
We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.
References:
- 1.
- H. Cohen: A Course In Computational Algebraic Number Theory. Springer-Verlag, Berlin et al. (Grad. Texts in Math. 138), 1993. MR 94i:11105
- 2.
- V. Ennola: Cubic number fields with exceptional units. In Computational Number Theory, Proc. of the Colloquium held at Debrecen, September 1989. Ed. A. Peth\H{o}, M.E. Pohst, H.C. Williams, and H.G. Zimmer. Walter de Gruyter, 103-138, 1991. MR 93e:11131
- 3.
- G.H. Hardy and E.M. Wright: An introduction to the theory of numbers. Fifth edition. Oxford University Press, 1979. MR 81i:10002
- 4.
- M. Laurent, M. Mignotte and Yu. Nesterenko: Formes linéaires en deux logarithmes et déterminants d'interpolation. J. Number Theory, Vol. 55, 285-321, 1995. MR 96h:11073
- 5.
- H.W. Lenstra Jr.: Euclidean number fields of large degree. Invent. Math., Vol. 38, 237-254, 1977. MR 55:2836
- 6.
- A. Leutbecher and G. Niklasch.: On cliques of exceptional units and Lenstra's construction of euclidean fields. In Number Theory, Proc Jour. Arith., Ulm 1987. Ed. H.P. Schlickewei and E. Wirsing, 150-178. Springer-Verlag, LNM 1380, 1989. MR 90i:11123
- 7.
- M. Mignotte: Verification of a conjecture of E. Thomas. J. Number Theory, Vol. 44, 172-177, 1993. MR 94m:11035
- 8.
- M. Mignotte, A. Peth\H{o} and R. Ralf: Complete solutions of a family of quartic Thue and index form equations. Math. Comp., Vol. 65, 341-354, 1996. MR 96d:11034
- 9.
- T. Nagell: Sur une propriété des unités d'un corps algébrique. Arkiv f. Matem., Vol. 5, 343-356, 1964. MR 32:7542
-: Sur les unités dans les corps biquadratiques primitifs du premier rang. loc. cit. Vol. 7, 359-394, 1968.MR 39:5511
-: Quelques problèmes relatifs aux unités algébriques. loc. cit. Vol. 8, 115-127, 1969.MR 42:3053
-: Sur un type particulier d'unités algébriques. loc. cit. Vol. 8, 163-184, 1969. MR 42:3064 - 10.
- G. Niklasch: Family portraits of exceptional units. Preprint MPI 95-117, Max-Planck-Inst. f. Math., Bonn, 1995.
- 11.
- A. Peth\H{o}: Complete solutions to families of quartic Thue equations. Math. Comp., Vol. 57, 777-798, 1991. MR 92e:11023
- 12.
- N.P. Smart: The solution of triangularly connected decomposable form equations. Math. Comp., Vol. 64, 819-840, 1995. MR 95f:11110
- 13.
- N.P. Smart: Solving Discriminant Form Equations via Unit Equations. J. Symbolic Computation, Vol. 21, 367-374, 1996. MR 97g:11029
- 14.
- E. Thomas: Complete solutions to a family of cubic diophantine equations. J. Number Theory, Vol. 34, 235-250, 1990. MR 91b:11027
- 15.
- N. Tzanakis and B.M.M. de Weger: On the practical solution of the Thue equation. J. Number Theory, Vol. 31, 99-132, 1989. MR 90c:11018
- 16.
- N. Tzanakis and B.M.M. de Weger: How to explicitly solve a Thue-Mahler equation. Comp. Math., Vol. 84, 223-288, 1992. MR 93k:11025
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Additional Information:
G.
Niklasch
Affiliation:
Zentrum Mathematik der TU / SCM, Technische Universität München, D--80290 München, Germany
Email:
nikl@mathematik.tu-muenchen.de
N.
P.
Smart
Affiliation:
Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, England
Address at time of publication:
Hewlett-Packard Laboratories, Fitton Road, Stoke Gifford, Bristol, BS12 6QZ, United Kingdom
Email:
N.P.Smart@ukc.ac.uk, nsma@hplb.hpl.hp.com
DOI:
10.1090/S0025-5718-98-00958-2
PII:
S 0025-5718(98)00958-2
Keywords:
Exceptional units,
Baker's method,
diophantine approximation
Received by editor(s):
October 18, 1996
Copyright of article:
Copyright
1998,
American Mathematical Society
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