Swan-like results for binomials and trinomials over finite fields of odd characteristic

摘要

Swan (Pac. J. Math. 12:1099-1106, 1962) gives conditions under which the trinomial x~n + x~k + 1 over F_2 is reducible. Vishne (Finite Fields Appl. 3:370-377, 1997) extends this result to trinomials over extensions of F_2. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field F_q, where q is a power of an odd prime.