The determination of Galois groups

Stauduhar, Richard P.

doi:10.1090/S0025-5718-1973-0327712-4

The determination of Galois groups
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by Richard P. Stauduhar PDF

Math. Comp. 27 (1973), 981-996 Request permission

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 $N \leqq 7$. Two exemplary calculations are given.

References

John Brillhart, On the Euler and Bernoulli polynomials, J. Reine Angew. Math. 234 (1969), 45–64. MR 242790, DOI 10.1515/crll.1969.234.45

Theory of Groups of Finite Order

Quart. J. Pure Appl. Math.

F. N. Cole, Note on the substitution groups of six, seven, and eight letters, Bull. Amer. Math. Soc. 2 (1893), no. 8, 184–190. MR 1557242, DOI 10.1090/S0002-9904-1893-00137-7
Edgar Dehn, Algebraic equations: An introduction to the theories of Lagrange and Galois, Dover Publications, Inc., New York, 1960. MR 0115991
G. A. Miller, Note on substitution groups of eight letters, Bull. Amer. Math. Soc. 3 (1894), no. 7, 168–169. MR 1557324, DOI 10.1090/S0002-9904-1894-00205-5
G. A. Miller, Note on the substitution groups of eight and nine letters, Bull. Amer. Math. Soc. 3 (1894), no. 10, 242–245. MR 1557348, DOI 10.1090/S0002-9904-1894-00223-7

Bull. Amer. Math. Soc.

Modern Algebra