What is the isomorphism between the fields $(Z_2[x]^{<3},+_{x^3+x^2+1},\times_{x^3+x^2+1})$ and $(Z_2[x]^{<3},+_{x^3+x+1},\times_{x^3+x+1})$?

Question

They are both Galois fields of order 8. I'm not exactly sure what the question means - how does one determine/describe an isomorphism?

Answer 1

These are terrible notations for the fields $\mathbb{F}_2[x]/(x^3+x^2+1)$ and $\mathbb{F}_2[x]/(x^3+x+1)$. Since we have $(x+1)^3+(x+1)+1=x^3+x^2+1$, the isomorphism $\mathbb{F}_2[x] \to \mathbb{F}_2[x]$, $x \mapsto x+1$ maps $(x^3+x+1)$ onto $(x^3+x^2+1)$, so that it induces an isomorphism $\mathbb{F}_2[x]/(x^3+x+1) \to \mathbb{F}_2[x]/(x^3+x^2+1)$.