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Solving norm equations in relative number fields using $S$-units

Author: Denis Simon
Journal: Math. Comp. 71 (2002), 1287-1305
MSC (2000): Primary 11D57, 11Y50, 11R29
Published electronically: January 11, 2002
MathSciNet review: 1898758
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Abstract: In this paper, we are interested in solving the so-called norm equation ${\mathcal N}_{L/K} (x)=a$, where $L/K$ is a given arbitrary extension of number fields and $a$ a given algebraic number of $K$. By considering $S$-units and relative class groups, we show that if there exists at least one solution (in $L$, but not necessarily in ${\mathbb Z}_L$), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the $S$-units that are norms are norms of $S$-units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When $a$ is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.

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  • [1] H.J. Bartels: Über Normen algebraischer Zahlen, Math. Ann., 251 (1980) 191-212. MR 81k:12010
  • [2] K. Brown: Cohomology of groups, Graduate Texts in Math., Vol. 87, Springer-Verlag (1982). MR 83k:20002
  • [3] C. Chevalley: Sur la théorie du corps de classe dans les corps finis et les corps locaux, J. Fac. Sci Tokyo, 2 (1933) 365-475.
  • [4] H. Cohen: A course in computational algebraic number theory, Graduate Texts in Math., Vol. 138, Springer-Verlag (1993). MR 94i:11105
  • [5] H. Cohen, F. Diaz y Diaz, M. Olivier: Computation of relative quadratic class groups, ANTS III, Springer LN in Computer Science, 1423 (J. Buhler Ed. 1998), p 433-440. MR 2000j:11165
  • [6] H. Cohen, F. Diaz y Diaz, M. Olivier: Algorithms for finite abelian groups, submitted to J. Symb. Comp.
  • [7] C. Fieker : Ueber Relative Normgleichungen in Algebraischen Zahlkörpern, Dissertation, Technische Univertität Berlin (1997);
  • [8] C. Fieker, A. Jurk, M. Pohst: On solving relative norm equations in algebraic number fields, Math. Comp., 66 (1997) 399-410. MR 97c:11118
  • [9] U. Fincke, M. Pohst: A procedure for determining algebraic integers of given norm, Proceedings EUROCAL 83, Springer LN in Computer Science, 162 (1983) 194-202. MR 86k:11078
  • [10] D. Garbanati: An algorithm for finding an algebraic number whose norm is a given rational number, J. Reine Angew. Math., 316 (1980) 1-13. MR 81k:12004
  • [11] J.-P. Serre: Corps Locaux, Hermann, 2ème éd. (1968). MR 50:7096
  • [12] C.L. Siegel: Normen algebraischer Zahlen, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II 1973, 197-215. MR 49:7237
  • [13] D. Simon: Équations dans les Corps de Nombres et Discriminants Minimaux, thèse, Université de Bordeaux I (1998).

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Additional Information

Denis Simon
Affiliation: Université Bordeaux I, Laboratoire A2X, 351 Cours de la Libération, 33405 Talence, France

Keywords: Relative number fields, norm equation, $S$-unit, class group
Received by editor(s): January 22, 1999
Received by editor(s) in revised form: April 13, 1999
Published electronically: January 11, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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