Find isomorphism between $\mathbb F_2[x]/(x^3+x+1)$ and $\mathbb F_2[x]/(x^3+x^2+1)$.

Question

Find isomorphism between $\mathbb F_2[x]/(x^3+x+1)$ and $\mathbb F_2[x]/(x^3+x^2+1)$.

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It is easy to construct an injection $f$ satisfying $f(a+b)=f(a)+f(b)$ and $f(ab)=f(a)f(b)$. However, I am stuck how to construct such a mapping that is bijective.

Thank you for help!

Answer 1

Note that if $y$ is a solution to $$y^3+y^2+1=0$$ then $y+1$ is a solution to $$x^3+x+1=0.$$

Answer 2

Note that $x^2=x (\mod 2) $

So $F_2[x]/(x^3+x+1) \simeq F[x]/(2,x^3+x+1) \simeq F[x]/(2,x^3+x^2+1) \simeq F_2[x]/ (x^3+x^2+1) $