The determination of Galois groups

Author:
Richard P. Stauduhar

Journal:
Math. Comp. **27** (1973), 981-996

MSC:
Primary 12-04; Secondary 12A55

DOI:
https://doi.org/10.1090/S0025-5718-1973-0327712-4

MathSciNet review:
0327712

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Abstract: A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the polynomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear factors of which determine new orderings for the roots. Sequences of these resolvents isolate the Galois group of the polynomial. Machine implementation of the technique requires the use of multiple-precision integer and multiple-precision real and complex floating-point arithmetic. Using this technique, the writer has developed programs for the determination of the Galois groups of polynomials of degree . Two exemplary calculations are given.

**1.**J. Brillhart, "On the Euler and Bernoulli polynomials,"*J. Reine Angew. Math.*, v. 234, 1969, pp. 45-64. MR**0242790 (39:4117)****[1]**W. Burnside,*Theory of Groups of Finite Order*, Cambridge Univ. Press, London, 1897.**[2]**A. Cayley, "On the substitution groups for two, three,. . ., eight letters,"*Quart. J. Pure Appl. Math.*, v. 25, 1891, pp. 71-88, 137-155.**[3]**F. N. Cole, "Note on the substitution groups of six, seven and eight letters,"*Bull. New York Math. Soc.*, v. 2, 1893, pp. 184-190. MR**1557242****[4]**E. Dehn,*Algebraic Equations*, Columbia Univ. Press, New York; reprint, Dover, New York, 1960. MR**0115991 (22:6788)****[5]**G. A. Miller, "Note on substitution groups of eight letters,"*Bull. New York Math. Soc.*, v. 3, 1894, pp. 168-169. MR**1557324****[6]**G. A. Miller, "Note on the substitution groups of eight and nine letters,"*Bull. New York Math. Soc.*, v. 3, 1894, pp. 242-245. MR**1557348****[7]**G. A. Miller, "Note on Burnside's theory of groups,"*Bull. Amer. Math. Soc.*, v. 5, 1899, pp. 249-251.**[8]**B. van der Waerden,*Modern Algebra*. Vol. I, Ungar, New York, 1953.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0327712-4

Keywords:
Galois group algorithm,
resolvent equations

Article copyright:
© Copyright 1973
American Mathematical Society