@article{Harrington_Jones_2017, title={The irreducibility of power compositional sextic polynomials and their Galois groups}, volume={120}, url={https://www.mscand.dk/article/view/25850}, DOI={10.7146/math.scand.a-25850}, abstractNote={<p>We define a <em>power compositional sextic polynomial</em> 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.</p>}, number={2}, journal={MATHEMATICA SCANDINAVICA}, author={Harrington, Joshua and Jones, Lenny}, year={2017}, month={May}, pages={181–194} }