Reducible Polynomials in finite fields

Question

I am stuck on the following question. Verify that $x^5 + x + 1$ is reducible in $Z_2[x]$ and find its factors.

Help would be much appreciated whether it is the answers with how you did it or just the method as to how I can proceed.

Thanks

Answer 1

Note that $f(x)=x^5+x+1$ does not have any linear factors because $x=0,1 \in \mathbb{Z}_2$ are not the roots of this polynomial $f(x)$. Thus the only way it may factor non-trivially is $$x^5+x+1=a(x)b(x) \qquad \text{with } \text{deg }a(x)=2 \text{ and } \text{deg }b(x)=3.$$ Now look for irreducible polynomials of degree $2$ in $\mathbb{Z}_2[x]$. In all there are $4$ polynomial of degree $2$, namely $$x^2+x+1, x^2+1, x^2+x, x^2.$$ It is easy to see that only the first is irreducible.

So if at all $f(x)$ factors non-trivially then $a(x)=x^2+x+1$. Now try to divide $f$ by $a$ and see what happens.