find the inverse of $x^2 + x + 1$ In $\mathbb{F}_2[x]$ modulo $x^4 + x + 1$
find the inverse of $x^2 + x + 1$
not 100% sure but here what i have:
user euclid algorithm:
$x^4 + x + 1 = (x^3 + 1)(x + 1) + x$
$(x^3 + 1) = x * x * x + 1$
$1 = (x^3 + 1) - x * x * x  $

Note that:
$(x^{2}+x+1)(x^{2}+x+1)=x^{2}+x$
$(x^{2}+x)(x^{2}+x+1)=x^{4}+x=1$. To get this I used that $x^{4}+x+1=0$ and that the coefficients are in $\mathbb{F}_{2}$.
So the inverse is $(x^{2}+x+1)^{2}=x^{2}+x$.
