On necessary and sufficient conditions for the monogeneity of a certain class of polynomials

Abstract

Abstract Let f(x) ∈ ℤ[x] be monic and irreducible over ℚ, with deg(f) = n. Let K = ℚ(θ), where f(θ) = 0, and let ℤ K denote the ring of integers of K. We say f(x) is monogenic if {1, θ, θ 2, …, θ n−1} is a basis for ℤ K . Otherwise, f(x) is called non-monogenic. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes p ≥ 3, and integers t ≥ 1 coprime to p, such that f(x) = x p − 2ptx p−1 + p 2 t 2 x p−2 + 1 is non-monogenic. Finally, we illustrate this situation with an explicit example.