arXiv2026
Constructing irreducible polynomials over finite fields via rational transformations is an active field of study. In this paper, we show that the method can be used for more than that. By considering normalized rational functions $Q$ that generate an invariant subfield $\mathbb{F}_q(x)^G$ of $\mathbb{F}_q(x)$, for a subgroup $G\le Aut(\mathbb{F}_q(x)/\mathbb{F}_q)$, and exploiting the factorization pattern of their rational transformations, we are able to prove that the sets arising from considering whether the rational transformation $f^Q$ has exactly $deg(Q)$ irreducible factors of the same degree as $f$, or an irreducible factor of higher degree, admit a recursive behavior, i.e. their numbers satisfy a recursive equation. These sets occasionally can be described in terms of arithmetical properties the roots of the irreducible polynomials satisfy, e.g. their roots satisfy certain trace or power/non-power conditions. These can often be translated to properties of their minimal polynomials. With that, we prove a new theorem about the number of certain irreducible factors of a class of linearized polynomials, which gives a new context to a recent paper by Granger.