Ytukyg

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 $4$.

It is easy to see that $x^2+x+1$ is irreducible since it has degree $2$ and it has no root. Hence the quotient ring is necessarily a field. Since the quotient has $4$ elements and a finite field is uniquely determined by the number of its elements, you get $mathbb{F}_2[x]/(x^2+x+1)congmathbb{F}_4$.