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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!

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  • $\begingroup$ What injection did you come up with? If it is actually additive then it should take 0 to 0... $\endgroup$ – TomGrubb Apr 23 '17 at 2:51
  • $\begingroup$ @ThomasGrubb sorry, I did not make it clear. I mean the problem lies in satisfying one-one, not taking 0 to 0. $\endgroup$ – Ivon Apr 23 '17 at 2:53
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    $\begingroup$ An injection on finite sets of the same cardinality is a bijection so you should be good then! $\endgroup$ – TomGrubb Apr 23 '17 at 2:54
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    $\begingroup$ If $\phi$ is a (non-zero) ring morphism $K \to F$ and $K$ is a field, then $\phi$ is a field isomorphism $K \to \phi(K)$. $\endgroup$ – reuns Apr 23 '17 at 2:58
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    $\begingroup$ You want a field isomorphism, I believe. $\endgroup$ – Lubin Apr 23 '17 at 3:15
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Note that if $y$ is a solution to $$y^3+y^2+1=0$$ then $y+1$ is a solution to $$x^3+x+1=0.$$

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    $\begingroup$ Funny, I would have said that if $y$, then $1/y$. $\endgroup$ – Lubin Apr 23 '17 at 3:15
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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) $

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