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The irreducibility of power compositional sextic polynomials and their Galois groups

Joshua Harrington, Lenny Jones


We define a power compositional sextic polynomial to be a monic sextic polynomial $f(x):=h(x^d)\in \mathbb{Z} [x]$, where $h(x)$ is an irreducible quadratic or cubic polynomial, and $d=3$ or $d=2$, respectively. In this article, we use a theorem of Capelli to give necessary and sufficient conditions for the reducibility of $f(x)$, and also a description of the factorization of $f(x)$ into irreducibles when $f(x)$ is reducible. In certain situations, when $f(x)$ is irreducible, we also give a simple algorithm to determine the Galois group of $f(x)$ without the calculation of resolvents. The algorithm requires only the use of the Rational Root Test and the calculation of a single discriminant. In addition, in each of these situations, we give infinite families of polynomials having the possible Galois groups.

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Bergé, A.-M., Martinet, J., and Olivier, M., The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), no. 190, 869–884.

Brown, S. C., On the galois groups of sextic trinomials, Master's thesis, University of British Columbia, 2011, url

Butler, G. and McKay, J., The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911.

Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993.

Eloff, D., Spearman, B. K., and Williams, K. S., $A_4$-sextic fields with a power basis, Missouri J. Math. Sci. 19 (2007), no. 3, 188–194.

Ide, J. and Jones, L., Infinite families of $A_4$-sextic polynomials, Canad. Math. Bull. 57 (2014), no. 3, 538–545.

Lavallee, M. J., Spearman, B. K., and Williams, K. S., Lifting monogenic cubic fields to monogenic sextic fields, Kodai Math. J. 34 (2011), no. 3, 410–425.

Olivier, M., The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comp. 58 (1992), no. 197, 419–432.

Schinzel, A., Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000.



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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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