Prove that the ideal $(x^3 + x + 1)$ in the polynomial ring $Z/(2)[x]$ over $Z/(2)$ is a prime ideal.

Question

There are two ways to prove this and in both ways I have not been able to: Let $A,B$ ideals in $Z/(2)[x]$ if $AB\subseteq (x^3 + x + 1)$ and $B\nsubseteq(x^3 + x + 1)$ I do not know how to prove that $A\subseteq (x^3 + x + 1)$. The other way is by proving that $Z/(2)[x]/(x^3 + x + 1)$ is simple, for it let $C+(x^3 + x + 1)$ ideals in $Z/(2)[x]/(x^3 + x + 1)$ and $C+(x^3 + x + 1)\neq (0)$ How can I prove that $C+(x^3 + x + 1)=Z/(2)[x]/(x^3 + x + 1)$ ?

Answer 1

The simpler way is to prove that $x^3 + x + 1$ is irreducible mod $2$, and this is easy because it does not have a root.