On the monogenity of composed polynomials

Abstract

Let (Formula presented.) be a monic irreducible polynomial of degree n. We say that (Formula presented.) is monogenic if, for a root (Formula presented.) of (Formula presented.), the set (Formula presented.) forms an integral basis of the ring of integers (Formula presented.) of the number field (Formula presented.). Consider (Formula presented.) with (Formula presented.), and (Formula presented.), where (Formula presented.) and (Formula presented.), such that (Formula presented.) is irreducible over (Formula presented.). In this study, we establish necessary and sufficient conditions involving a, b, c, d, m, n for the polynomial (Formula presented.) to be monogenic. Additionally, we examine the nature of solutions to specific differential equations, and present a class of monogenic polynomials with non-square-free discriminants as an application.