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

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

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!

• What injection did you come up with? If it is actually additive then it should take 0 to 0... – TomGrubb Apr 23 '17 at 2:51
• @ThomasGrubb sorry, I did not make it clear. I mean the problem lies in satisfying one-one, not taking 0 to 0. – Ivon Apr 23 '17 at 2:53
• An injection on finite sets of the same cardinality is a bijection so you should be good then! – TomGrubb Apr 23 '17 at 2:54
• If $\phi$ is a (non-zero) ring morphism $K \to F$ and $K$ is a field, then $\phi$ is a field isomorphism $K \to \phi(K)$. – reuns Apr 23 '17 at 2:58
• You want a field isomorphism, I believe. – Lubin Apr 23 '17 at 3:15

Note that if $y$ is a solution to $$y^3+y^2+1=0$$ then $y+1$ is a solution to $$x^3+x+1=0.$$
• Funny, I would have said that if $y$, then $1/y$. – Lubin Apr 23 '17 at 3:15
Note that $x^2=x (\mod 2)$
So $F_2[x]/(x^3+x+1) \simeq F[x]/(2,x^3+x+1) \simeq F[x]/(2,x^3+x^2+1) \simeq F_2[x]/ (x^3+x^2+1)$