Here is a question from an old exam:
Show that there are infinite $n\in \mathbf{N}, A= x^{n}+x+1 $ which are reducible over $\mathbf{F}_{2}[x]$.
Using André Nicolas' and Qiaochu Yuan's hint: $x^{2}+x+1$ as dividing polynomial. $x^{2}+x+1$ is irreducible over $\mathbf{F_{2}}$. If an irreducible polynomial divides another polynomial which is not itself, that means that polynomial must be reducible. We want to show that $x^{2}+x+1$ divides all polynomials of the form $x^{3n+5}+x+1$. I can't figure the induction steps, but in $\mathbf{F_{2}}$ the polynomial belongs to the residue class $\tilde{1}$, therefore there must be an infnite amount of them.
Concerning Gerry Myerson's hint, how can I use cubic roots in $\mathbf{F_{2}}$, wouldn't I need $\mathbf{R}[i]$ for that?
Help is greatly appreciated.