A note on the reducibility of binary affine polynomials

Abstract

Stickelberger---Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over $${\mathbb{F}_2}$$ with degree >1, where A(x) = L(x) + 1 and $${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$$ is a linearized polynomial over $${\mathbb{F}_2}$$ , is reducible except x 2 + x + 1 and x 4 + x + 1. We also give some explicit factors of some special affine pentanomials over $${\mathbb{F}_2}$$ .