Factorization of Trinomials over Galois Fields of Characteristic 2

We study the parity of the number of irreducible factors of trinomials over Galois fields of characteristic 2. As a consequence, some sufficient conditions for a trinomial being reducible are obtained. For example,xn+axk+b GF(2t)[x] is reducible if bothn,tare even, except possibly whenn= 2k,kodd. The caset= 1 was treated by R. G. Swan (Pacific J. Math.12,No. 2 (1962), 1099–1106), who showed thatxn+xk+ 1 is reducible overGF(2) if 8n.