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.

More permutation polynomials with Niho exponents which permute Fq2 Abstract   Constructions of permutation polynomials over finite fields have attracted much interests in recent years, especially those with few terms, such as trinomials, due to their simple form and additional properties. In this paper, we construct several classes of permutation trinomials over      F      p    2  k        with Niho exponents of the form    f  (  x  )    x      xcexbb    1      x    s  (    p    k    xe2x88x92  1  )    1        xcexbb    2      x    t  (    p    k    xe2x88x92  1  )    1     ; some necessary and sufficient conditions for the polynomial    f  (  x  )    to permute      F      p    2  k        are provided. Specifically, for    p    5   , new permutation trinomials are presented. We also give recursive constructions of permutation polynomials using selfreciprocal polynomials.