A Note on Stable Quadratic Polynomials over Fields of Characteristic Two

TL;DR

This paper investigates the stability of quadratic polynomials over fields of characteristic two, proving non-existence over finite fields and existence over certain function fields, thereby clarifying their behavior in different field contexts.

Contribution

It establishes the non-existence of stable quadratic polynomials over finite fields of characteristic two and demonstrates their existence over some function fields of the same characteristic.

Findings

No stable quadratic polynomials over finite fields of characteristic two

Existence of stable quadratic polynomials over some function fields of characteristic two

Clarifies stability behavior in different field types

Abstract

In this note, first we show that there is no stable quadratic polynomial over finite fields of characteristic two and then show that there exist stable quadratic polynomials over function fields of characteristic two.