What question did this study set out to answer?

This research aims to define conditions under which iterates of an irreducible polynomial are monogenic in a number field.

June 27, 2026Open Access

Critical point criteria and dynamically monogenic polynomials

Key Points

This research aims to define conditions under which iterates of an irreducible polynomial are monogenic in a number field.
Analyzed critical points of a monic irreducible polynomial f(x) over a number field K.
Formulated necessary and sufficient conditions for backwards orbits to yield monogenic elements.
Constructed examples illustrating infinite towers of monogenic fields via identified conditions.
Demonstrated that certain critical points determine the nature of iterates being monogenic.
Provided concrete examples of dynamically monogenic polynomials leading to infinitely many monogenic number fields.
Revealed that the backward orbit of one monogenerator can produce a new monogenerator at the subsequent level.

Abstract

Let K be a number field with ring of integers OK, and let f (x) Kx be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of f (x) for the iterates of f (x) to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backwards orbits of elements of OK under f (x) to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level.

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