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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.


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References


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DOI: http://dx.doi.org/10.7146/math.scand.a-25850

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

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