A Note on the Monogeneity of Power Maps

Let φ(x)=xd−t∈ℤ[x] be an irreducible polynomial of degree d≥2, and let θ be a root of φ. The purpose of this paper is to establish necessary and sufficient conditions for φ(x) to be monogenic, meaning the ring of integers of ℚ(θ) is generated by the powers of a root of φ(x). Sufficient conditions for monogeneity are established using Dedekind’s criterion. We then apply the Montes algorithm to give an explicit formula for the discriminant of ℚ(θ). Together, these results can be used to determine when φ(x) is not monogenic.