A note on the stability of trinomials over finite fields

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. Specially, we show that if $f(x)$ is a trinomial of even degree over the binary field $\mathbb{F}_2$, then $f(x)$ is not stable. We prove a similar result for some families of monic trinomials over finite fields of odd characteristic. These results are obtained towards the resolution of a conjecture on the instability of polynomials over finite fields whose degrees are divisible by the characteristic of the underlying field.

[1]  K. Dalen.  On a theorem of Stickelberger , 1955 .

[2]  Domingo Gómez-Pérez,et al.  An estimate on the number of stable quadratic polynomials , 2010, Finite Fields Their Appl..

[3]  N. Boston,et al.  Settled polynomials over finite fields , 2012 .

[4]  Donald L. Mcquillan,et al.  Irreducibility of the iterates of a quadratic polynomial over a field , 2000 .

[5]  J. Silverman.  The Arithmetic of Dynamical Systems , 2007 .

[6]  Daniel Panario,et al.  Swan-like results for binomials and trinomials over finite fields of odd characteristic , 2011, Des. Codes Cryptogr..

[7]  F. Luca,et al.  ON STABLE QUADRATIC POLYNOMIALS , 2012, Glasgow Mathematical Journal.

[8]  Stabilité des polynômes , 2005 .

[9]  A. Nicolás,et al.  Stable polynomials over finite fields , 2012, 1206.4979.

[10]  R. G. Swan,et al.  Factorization of polynomials over finite fields. , 1962 .

[11]  Rudolf Lide,et al.  Finite fields , 1983 .

[12]  Nigel Boston,et al.  A Refined Conjecture for Factorizations of Iterates of Quadratic Polynomials over Finite Fields , 2015, Exp. Math..

[13]  Omran Ahmadi,et al.  The trace spectra of polynomial bases for $${\mathbb{F}}_{2^{n}}$$ , 2007, Applicable Algebra in Engineering, Communication and Computing.

[14]  Giacomo Micheli,et al.  Irreducible polynomials over finite fields produced by composition of quadratics , 2019, Revista Matemática Iberoamericana.

[15]  Stephen D. Cohen.  On irreducible polynomials of certain types in finite fields , 1969 .

[16]  R. W. K. Odoni.  The Galois Theory of Iterates and Composites of Polynomials , 1985 .