A note on the stability of trinomials over finite fields

Abstract A polynomial f ( x ) over a field K is called stable if all of its iterates are irreducible over K. In this paper, we study the stability of trinomials over finite fields. We show that if f ( x ) is a trinomial of even degree over the binary field F 2 , then f ( x ) is not stable. We prove similar results for some families of monic trinomials over finite fields of odd characteristic. We also study the stability of polynomials of higher weights and prove some results and pose a new conjecture.