On The Index of Polynomial Compositions over Valued Fields

Abstract: Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions $f(h(x))$, where $f(x)$ and $h(x)$ are monic polynomials over an arbitrary Krull valuation ring and $h(x)$ is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when $f(h(x))$ generates a monogenic field. In particular, we relate the monogeneity of the composition to that of $f(x)$ and to explicit square-free conditions on the critical values of $h(x)$. We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.