The irreducibility of power compositional sextic polynomials and their Galois groups - Harrington, Joshua; Jones, Lenny (Mathematica Scandinavica, 2017)
We define a power compositional sextic polynomial to be a monic sextic polynomial f(x):=h(x^d)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.