prove $\Bbb{Z}_2[x] /(x^3+x+1)$ is a field Prove that $$\Bbb{Z}_2[x] /(x^3+x+1)$$ is a field. 

Attempt 1
$\Bbb{Z}_2 $ is a field , $p(x)=x^3+x+1$ is a non constant polynomial in $\Bbb{Z}_2[x]$
$(x^3+x+1)$ is irreducible in $\Bbb{Z}_2[x]$ then $\Bbb{Z}_2[x]/\big(p(x)\big)$ is a field. 
it has been a while since taking ring theory appreciate corrections and/or ciriticism

There are a few ways to show this. Theoretically, $x^3+x+1$ is irreducible, and so it is a maximal ideal in $k[x]$ so the quotient is a field. This is probably the quickest way. To show that it is irreducible, note that if $x^3+x+1$ were reducible, it would factor into the product of three linear components or one linear and one quadratic.
Thus, one only needs to show that  roots of $x^3+x+1 $ are not in $\mathbb Z_2$. In particular, if $x^3+x+1=0$, then $x^3+x=1$, so, but this is impossible, since $x=0,1$ both give $0$ as solutions.
