Proof that the Galois Field of order 8 is a field. We know that the Galois Field of order 8 is isomorphic with $$\left( Z_2[x]^{< 3}, +_{d(x)}, \times_{d(x)}\right) $$ (Field of polynomials with coefficients in $Z_2$ and of grade smaller than 3, and $+_{d(x)}$ and $\times_{d(x)}$ being the sum and multiplication $mod(d(x))$ )
Is it thus sufficient to prove that $\left( Z_2[x]^{< 3}, +_{d(x)}, \times_{d(x)}\right) $ is a field? Is there a better way?
Subquestion: Which polynomial d(x) should I consider? Since there are two possible here: $x^3+x^2+1$ and $x^3+x+1$.

Your notation is a little confusing but I guess I can answer part of your question at least. 
Every finite field of same order is isomorphic to each other so it is indeed sufficient to construct a field of order 8 to know exactly how the field operates.
Since $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}:=\mathbb{F}_2$ is a field it is sufficient to obtain an irreducible polynomial of degree 3 to construct a field of order 8. Either of the polynomial you mentioned are irreducible so 
$$\mathbb{F}_2[x]/d(x)$$ where $d(x)$ can be chosen to be any of the two polynomial you mentioned, is a field of order 8. They will be isomorphic but the isomorphism won't necessarily be easy to construct.
