$\mathbb{F}_2[X]/(S) \cong \mathbb{F}_4 $ Let $S(X) = X^2 +X+1 \in \mathbb{F}_2[X]$
Prove that $\mathbb{F}_2[X]/(S) \cong \mathbb{F}_4 $
What I did:
$\{1, X \}$ is a basis of $\mathbb{F}_2[X]/(S)$ and S is irreducible in $\mathbb{F}_2$ so $ |\mathbb{F}_2[X]/(S)| = 2^2 = 4$
I need help to prove that  $\mathbb{F}_2[X]/(S) \cong \mathbb{F}_4 $. 
Thank you.

What Trevor Gunn said is right. The statement you want to prove is not correct. First you need to show that $S(X) = X^5 + X^2 + 1$ is irreducible over $\mathbb{F}_2$. Then it will follow that $F[X]/\langle S \rangle$ is a field spanned by the elements $1, X, X^2, X^3, X^4$ and is thus isomorphic to $\mathbb{F}_{32}$ (remember that there is exactly one field with 32 elements).
Edit: OP has now changed the polynomial in question to $S(X) = X^2 + X + 1$. In this case the quotient ring $F[X]/\langle S(X) \rangle$ is a field spanned by the elements $1, X$ and thus has cardinality $2^2 = 4$, as mentioned in the question. The identification with $\mathbb{F}_4$ then comes from the fact that, up to isomorphism, there is only one field of cardinality $4$.
