Partial solvability by radicals

ABSTRACT

This paper is devoted to show, first, how to easily determine, when it exists, a non-trivial element in the centre of the Galois group of an irreducible polynomial in ℤ[x] and, second, how to deal in an efficient way with solvability by radicals when a non-trivial element in the centre of the Galois group of the considered polynomial is available.

REFERENCES

• 1: Vincenzo Acciaro , Jürgen Klüners, Computing automorphisms of abelian number fields, Mathematics of Computation, v.68 n.227, p.1179-1186, July 1999 [doi>10.1090/S0025-5718-99-01084-4]
• 2: Eric Bach , Jonathan Sorenson, Explicit bounds for primes in residue classes, Mathematics of Computation, v.65 n.216, p.1717-1735, Oct. 1996 [doi>10.1090/S0025-5718-96-00763-6]
• 3: J. W. S. Cassels, Local fields, London Math. Soc., Student Texts 3, Cambridge University Press, 1986.
• 4: George E. Collins , Mark J. Encarnación, Efficient rational number reconstruction, Journal of Symbolic Computation, v.20 n.3, p.287-297, Sept. 1995 [doi>10.1006/jsco.1995.1051]
• 5: H. Darmon, D. Ford, Computational verification of M11and M12as Galois group over ℚ, Comm. Algebra 17, 2941-2943, 1989.
• 6: J. D. Dixon, Exact solution of linear equations using p-adic expansions, Numer. Math. 40, 137-141, 1982.
• 7: J. D. Dixon, Computing subfields in algebraic number fields, J. Austral. Math. Society 49, 434-448, 1990.
• 8: P. Fernandez-Ferreiros, M. A. Gomez-Molleda, Computing the centre of the Galois group, submitted, 2001.
• 9: L. Gaal, Classical Galois theory with examples, Second edition, Chelsea Publishing Co., New York, 1973.
• 10: The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.2; Aachen, St Andrews, 1999. (http://www-gap.dcs.st-and.ac.uk/~ gap)
• 11: G. Hanrot , F. Morain, Solvability by radicals from an algorithmic point of view, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.175-182, July 2001, London, Ontario, Canada [doi>10.1145/384101.384125]
• 12: B. Huppert,Endliche Gruppen I, Springer-Verlag, 1967.
• 13: Jürgen Klüners , Gunter Malle, Explicit Galois realization of transitive groups of degree up to 15, Journal of Symbolic Computation, v.30 n.6, p.675-716, Dec. 2000 [doi>10.1006/jsco.2000.0378]
• 14: J. C. Lagarias, A. M. Odlyzko, Effective version of the Chebotarev density theorem, in Algebraic number fields (L-functions and Galois properties), A. Fröhlich ed., pp. 409-464, Academic Press, London, 1977.
• 15: S. Landau, Factoring polynomials over algebraic number fields, SIAM J. Comput. 14, 184-195, 1985.
• 16: S. Landau, G. L. Miller, Solvability by radicals is in polynomial time, J. Comput. System Sci. 30, 179-208, 1985.
• 17: G. Malle, B. H. Matzat, Inverse Galois Theory, Springer Monographs in mathematics, 2000.
• 18: B. L. van der Waerden, Modern Algebra, vol I, 1953.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

General Terms:
Standardization, Theory