Mathematics of Computation

Author(s): Thomas Mattman; John McKay.
Journal: Math. Comp. 66 (1997), 823-831.
MSC (1991): Primary 12F10, 12Y05
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Abstract: Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over $\mathbb Q (t_1,t_2,\ldots ,t_m)$ in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

1.: G. Butler and J. McKay, `The transitive groups of degree up to 11', Comm. Algebra 11 (1983), 863-911. MR 84f:20005
2.: D. Casperson and J. McKay, `Symmetric functions, -sets, and Galois groups', Math. Comp. Vol. 63, No. 208 (1994), 749-757. MR 95a:12001
3.: -, `An ideal decomposition algorithm', preliminary report, AMS Abstracts 13 (1992) 405.
4.: H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993 (ISBN 3-540-55640-0). MR 94i:11105
5.: J. Conway, A. Hulpke, J. McKay, ` On transitive permutation groups' (to appear).
6.: H. Darmon, private communication (1986).
7.: H. Darmon and D. Ford, `Computational verification of $M_{11}$ and $M_{12}$ as Galois groups over $\mathbb Q$ ', Comm. Alg. 17 (1989), 2941-2943. MR 91b:11146
8.: F-P. Heider and P. Kolvenbach, `The construction of SL(2,3)-polynomials', J. Number Theory 19 (1984) 392-411. MR 86g:11063
9.: A. Hulpke, Konstruktion transitiver Permutationsgruppen. PhD. thesis, RWTH-Aachen, Aachen, Germany, 1996.
10.: T. W. Mattman, The computation of Galois groups over function fields. Master's thesis, McGill University, Montréal, Québec, Canada, December 1992.
11.: B. H. Matzat, `Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe', J. Reine und Angew. Math. 349 (1984), 179-220. MR 85j:11164
12.: J. McKay, `Advances in computational Galois theory', in Computers in algebra, (Martin C. Tangora, Ed.), Lecture notes in pure and applied mathematics., vol. 111, pp.99-101, (1988). CMP 90:14
13.: J. McKay and E. Regener, `Actions of permutation groups on r-sets', Comm. Algebra 13 (1985) 619-630. MR 86j:20004
14.: E.A. O'Brien, `The groups of order 256', J. Algebra 143 (1991) 219-235. MR 93e:20029
15.: M. Olivier, Calcul des groupes de Galois de degré 8,9,10, et 11. Université Bordeaux I, 12 Oct. 1991 et 12 Fev. 1993.
16.: E. Skrzipczyk, Charaktertafeln von -Gruppen. Diplomarbeit, Lehrstuhl D fur Mathematik, Rheinisch-Westfalische Technische Hochschule Aachen, Aachen, Germany, 1992.
17.: L. Soicher, The computation of Galois groups. Master's thesis, University of Concordia, Montréal, Québec, Canada, April 1981.
18.: -, `An Algorithm for Computing Galois Groups', in Computational Group Theory (M. D. Atkinson, Ed.), pp. 291-296, Academic Press, 1984. MR 86d:12002b
19.: L. Soicher and J. McKay, `Computing Galois groups over the rationals', J. Number Theory 20 (1985), 273-281. MR 87a:12002
20.: Gene W. Smith, `Some Polynomials over $\mathbb Q (t)$ and their Galois groups.' (to appear in Math. Comp.).
21.: R. P. Stauduhar, The automatic determination of Galois groups. Ph. D. Dissertation, University of California, Berkeley, 1969.
22.: -, `The determination of Galois groups', Math. Comp. 27 (1973) 981-996. MR 48:6054
23.: S. Strelitz, `On the Routh-Hurwitz Problem', Amer. Math. Monthly 8 (1977) 542-544. MR 57:288
24.: B. van der Waerden, Modern Algebra. Vol. I, Ungar, (1949) Chapter 7.61. MR 10:587b

Thomas Mattman
Affiliation: Mathematics Department, McGill University, Montréal, H3A 2K6, Canada
Email: mattman@math.mcgill.ca

Yokoyama, K., A modular method for computing the Galois groups of polynomials., J. Pure Appl. Algebra 117/118 (1997), 617--636.. MR 98h:12005

Smith, Gene Ward , Some polynomials over $\bold Q(t)$ and their Galois groups., Math. Comp. 69 (2000), 775--796. MR 2000i:12007

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