Galois field isomorphism

Question

If α and β are roots of $x^3+x+1$ and$x^3+x^2+1$ ∈ $Z_2[x]$,respectively, prove that the Galois fields $Z_2(α)$ and $Z_2(β)$ are isomorphic.

I have no idea how to solve this problem. I've gone through all of the theorems in the chapter of the book that I am using, and there is nothing. Any help would be great, thank you in advance!

Hint: There is an isomorphism such that $\alpha\mapsto 1/\beta$. — Jyrki Lahtonen, May 07 '19 at 05:43
What is that isomorphism? It makes somewhat intuitive sense that there is one, but I'm not sure what it is. @JyrkiLahtonen — James Done, May 07 '19 at 05:58
A homomorphism of fields must map $1\mapsto 1$, and preserve sums and products. So if we know how it maps $\alpha$ we know how it must map $\alpha^2$, $1+\alpha$ etc. — Jyrki Lahtonen, May 07 '19 at 06:01
The key property is that $1/\beta$ is a zero of the minimal polynomial of $\alpha$. — Jyrki Lahtonen, May 07 '19 at 06:07

score 1 · Answer 1 · answered May 07 '19 at 05:32

Are you aware that for each prime $p$ and $n\in\Bbb Z^+$ there is a unique (up to isomorphism) field of order $p^n$? If so then you can apply this here; just note that the polynomials you've given are both irreducible over $\Bbb Z_2$ (why?), so $\Bbb Z_2[\alpha]$ and $\Bbb Z_2[\beta]$ both have degree $3$ over $\Bbb Z_2$, so they both have order $2^3=8$.

Galois field isomorphism

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