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

Question

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 02:51
@ThomasGrubb sorry, I did not make it clear. I mean the problem lies in satisfying one-one, not taking 0 to 0. — user771160, Apr 23 '17 at 02:53
An injection on finite sets of the same cardinality is a bijection so you should be good then! — TomGrubb, Apr 23 '17 at 02:54
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 02:58
@user1952009 I understand the principle you mention. Could you please explain how to construct the $\phi$? — user771160, Apr 23 '17 at 03:08

score 2 · Answer 1 · answered Apr 23 '17 at 03:13

2

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

answered Apr 23 '17 at 03:13

Rene Schipperus

40,478

3

Funny, I would have said that if $y$, then $1/y$. – Lubin Apr 23 '17 at 03:15

score 2 · Answer 2 · answered Apr 23 '17 at 06:19

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) $

answered Apr 23 '17 at 06:19

Suman Kundu

2,258

This answer is wrong: $x$ is an indeterminate, not an element of $\mathbb F_2$ or $\mathbb Z$. (Only the answerer knows what is $F$ in his answer.) – user26857 Nov 03 '21 at 20:09

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

2 Answers2