Stability of binomials over finite fields

TL;DR

This paper investigates the stability of binomials over finite fields, providing necessary and sufficient conditions for stability, and catalogs stable binomials of specific forms over small finite fields.

Contribution

It establishes that irreducible monic binomials are not always stable over finite fields and offers explicit criteria and a comprehensive list of stable binomials for small fields.

Findings

Stable binomials over finite fields are characterized by new criteria.

A table of stable binomials for fields with q ≤ 27 and degree d ≤ 10 is provided.

The work connects polynomial stability with properties of Mersenne primes.

Abstract

A polynomial $f(x)$ over a field $K$ is said to be stable if all its iterates are irreducible over $K$. L. Danielson and B. Fein have shown that over a large class of fields $K$, if $f(x)$ is an irreducible monic binomial, then it is stable over $K$. In this paper it is proved that this result no longer holds over finite fields. Necessary and sufficient conditions are given in order that a given binomial is stable over $\mathbb{F}_q$. These conditions are used to construct a table listing the stable binomials over $\mathbb{F}_q$ of the form $f(x)=x^d-a$, $a\in\mathbb{F}_q\setminus\{0,1\}$, for $q \leq 27$ and $d \leq 10$. The paper ends with a brief link with Mersenne primes.