A power compositional sextic polynomial is 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 the paper under review, by using a theorem of Capelli, the authors give a necessary and sufficient conditions for the irreducibility of \(f(x)\). Moreover, when \(f(x)\) is reducible, they describe the degree-type of the factorization of \(f(x)\) into irreducibles. In section 5, a simple algorithm to determine the irreducibility of \(f(x)\) and, without the use of resolvents, the galois group of \(f(x)\), when \(f(x)\) is irreducible. This algorithm requires only the use of the Rational Root Test.
The authors provide infinite families of polynomials having the possible Galois groups by application of the algorithm.