The irreducibility of power compositional sextic polynomials and their Galois groups

  • Joshua Harrington
  • Lenny Jones

Abstract

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.

References

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. https://doi.org/10.2307/2008516

Brown, S. C., On the galois groups of sextic trinomials, Master's thesis, University of British Columbia, 2011, url http://hdl.handle.net/2429/36998. https://doi.org/10.14288/1.0074288

Butler, G. and McKay, J., The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. https://doi.org/10.1080/00927878308822884

Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. https://doi.org/10.1007/978-3-662-02945-9

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. https://doi.org/10.4153/CMB-2014-008-1

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. https://doi.org/10.2996/kmj/1320935550

Olivier, M., The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comp. 58 (1992), no. 197, 419–432. https://doi.org/10.2307/2153044

Schinzel, A., Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511542916

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Published
2017-05-27
How to Cite
Harrington, J., & Jones, L. (2017). The irreducibility of power compositional sextic polynomials and their Galois groups. MATHEMATICA SCANDINAVICA, 120(2), 181-194. https://doi.org/10.7146/math.scand.a-25850
Section
Articles