How to show $\mathbb Z_2[x]/(x^3+x+1)$ is isomorphic to $\mathbb Z_2[x]/(x^3+x^2+1)$

Question

I am trying to figure out how to prove that $\mathbb Z_2[x]/(x^3+x^2+1)$ and $\mathbb Z_2[x]/(x^3+x+1)$ are isomorphic.

I know these sets all have the form $\{a+bx+cx^2\mid a,b,c\in\mathbb Z_2\}$. Is there any idea about it?

Answer 1

Since both the polynomials $(x^3+x+1)$ and $(x^3+x^2+1)$ are irreducible in $\mathbb{Z}_2[x]$, both $ \mathbb{Z}_2[x]/(x^3+x+1)$ and $\mathbb{Z}_2[x]/(x^3+x^2+1)$ are fields of order $8$ and are isomorphic.

Answer 2

$X\mapsto X+1$ is an automorphism of $\mathbb Z_2[X]$ which sends $X^3+X+1$ to $X^3+X^2+1$.