A Swan-like theorem

Richard G. Swan proved in 1962 that trinomials x8k+xm+1∈F2[x] with 8k>m have an even number of irreducible factors, and so cannot be irreducible. In fact, he found the parity of the number of irreducible factors for any square-free trinomial in F2[x]. We prove a result that is similar in spirit. Namely, suppose n is odd and , where . We show that if then f has an odd number of irreducible factors, and if then f has an even number of irreducible factors. This has an application to the problem of finding polynomial bases {1,α,…,αn-1} of F2n such that Tr(αi)=0 for all 1⩽i