A Study of monogenity of Binomial Composition

Let θ be a root of a monic polynomial h(x) ∈ [x] of degree n ≥ 2. We say $h(x)$ is monogenic if it is irreducible over and \ 1, θ, θ², …, θⁿ⁻¹ \ is a basis for the ring K of integers of K = (θ). In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup [θ] in K where K = (θ) with θ having minimal polynomial F(x) = (xᵐ-b)ⁿ - a ∈ [x], m≥ 1 and n ≥ 2. As an application, we provide a class of pairs of binomials f(x)=xⁿ-a and g(x)=xᵐ-b having the property that both $f(x)$ and $f(g(x))$ are monogenic.