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Finding Integral Linear Dependencies of Algebraic Numbers and Algebraic Lie Algebras

Published online by Cambridge University Press:  01 February 2010

Claus Fieker  and
Willem A. de Graaf
Claus Fieker
Affiliation:
School of Mathematics and Statistics, University of Sydney, Australia, claus@maths.usyd.edu.au, http://magma.maths.usyd.edu.au/users/claus
Willem A. de Graaf
Affiliation:
Dipartimento di Matematica, Università di Trento, Italy, degraaf@science.unitn.it, http://www.science.unitn.it/~degraaf/

Abstract

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Abstract:We give an algorithm for finding the module of linear dependencies of the roots of a monic integral polynomial. Using this, we describe an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 271 - 287
Copyright
Copyright © London Mathematical Society 2007

References

reference

1.Belabas, K., ‘A relative van Hoeij algorithm over number fields’, J. Symbolic Comput. 37 (2004) 641–668.Google Scholar
2.Borel, A., Linear algebraic groups, 2nd edn (Springer, Berlin/Heidelberg/ New York, 1991).CrossRefGoogle Scholar
3.Bosma, Wieb, Cannon, John J. and Playoust, Catherine, ‘The MAGMA algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235–266.CrossRefGoogle Scholar
4.Cannon, John J., ‘MAGMA’, http://magma.maths.usyd.edu.au.Google Scholar
5.Chevally, Claude, Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind. 1152 (Hermann & Cie., Paris, 1951).Google Scholar
6.Cohen, Arjeh M., Murray, Scott H. and Taylor, D. E., ‘Computing in groups of Lie type‘, Math. Comp. 73 (2004) 1477–1498 (electronic).Google Scholar
7.Cohen, Henri, A course in computational algebraic number theory, 1st edn, Graduate Texts in Mathematics 138 (Springer, Berlin, 1993).CrossRefGoogle Scholar
8.Derksen, Harm, Jeandel, Emmanuel and Koiran, Pascal, ‘Quantum automata and algebraic groups’, J. Symbolic Comput. 39 (2005) 357–371.CrossRefGoogle Scholar
9.Fieker, C. and Friedrichs, C., ‘On reconstruction of algebraic numbers’, Proceedings of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, 2-7 July, 2000, (ed. Bosma, W., Lecture Notes in Comput. Sci. (1838) (Springer, Berlin, (2000) 285–296.Google Scholar
10.Geissler, K., ‘Berechnung von Galoisgruppen über Zahl- und Funktionen- körpern’, PhD Thesis, TU-Berlin, 2003.Google Scholar
11.Grunewald, Fritz and Segal, Daniel, ‘Some general algorithms. I. Arithmetic groups’, Ann. of Math. (2) 112 (1980) 531–583.CrossRefGoogle Scholar
12.James, Gordon and Liebeck, Martin, Representations and characters of groups, 2nd edn (Cambridge University Press, New York, 2001).CrossRefGoogle Scholar
13.Just, Bettina, ‘Integer relations among algebraic numbers’, Math. Comp. 54 (1990) 467–477.Google Scholar
14.Klüners, J. and Malle, G., ‘A database for field extensions of the rationals‘, lms J. Comput. Math 4 (2001) 182–196, http://www.lms.ac.uk/jcm/4/lms2001-004.Google Scholar
15.Lang, Serge, Algebraic number theory, 2nd edn, Graduate Texts in Mathematics 110 (Springer, Berlin, 1994).CrossRefGoogle Scholar
16.MASSER, D. W., ‘Linear relations in algebraic groups’, New advances in transencendence theory (ed. Baker, Alan, Cambridge University Press, New York, 1988) 248–262.Google Scholar
17.Nguyen, P. and Stehlé, D., ‘Floating-point LLL revisited’, Proceedings of Eurocrypt (2005), Lecture Notes in Comput. Sci. 3494 (Springer, Berlin, (2005) 215–233.Google Scholar
18.Renault, Guéna El and Yokoyama, Kazuhiro, ‘A modular method for computing the splitting field of a polynomial’, Algorithmic Number Theory Symposium (ANTS VII), ed. Hess, Florian, Pauli, Sebastian and Pohst, Michael, Lecture Notes in Comput. Sci. 4076 (Springer, Berlin, 2006) 124–140.Google Scholar
19.Rotman, Joseph, Galois theory, 2nd edn, Universitext (Springer, New York, 1998).CrossRefGoogle Scholar
20.Schenkman, Eugene, Group theory (D. Van Nostrand Co., Inc., Princeton,N.J./Toronto, Ont./London, 1965).Google Scholar
21.Serre, Jean-Pierre, Local fields (Springer, Berlin, 1979).Google Scholar