The parity of the number of irreducible factors for some pentanomials (Q623239)

Using the Stickelberger-Swan theorem, the authors determine the parity of the number of irreducible factors for polynomials of the form \[ f(x)=x^m+x^{n+2}+x^{n+1}+x^n+1\in{\mathbb F}_2[x] \] where \(m\) is even and \(n<m-2\). In particular, they derive conditions (on \(m\) and \(n\)) for the existence of irreducible pentanomials over \({\mathbb F}_2\).