The irreducibility and monogenicity of power-compositional trinomials

A polynomial f(x) ∈ Z[x] of degree N is called monogenic if f(x) is irreducible over Q and {1, θ, θ2, . . . , θN−1} is a basis for the ring of integers of Q(θ), where f(θ) = 0. Define F(x) := xm+Axm−1+B. In this article, we determine sets of conditions on m, A, and B, such that the power-compositional trinomial F(xpn) is monogenic for all integers n ≥ 0 and a given prime p. Furthermore, we prove the actual existence of infinite families of such trinomials F(x).

Mathematics Subject Classification. Primary 11R04; Secondary 11R09, 12F05.