With the following field, $F=\mathbb{F}_2[y]/(y^3+y+1)$, I need to factorize $x^3+x+1$ and $x^3+x^2+1$ in the ring $F[x]$. I have done this and found out the roots of the first polynomial is $y,y^2,y^4$. Then the second is just the first polynomial with $x+1$ substituted into the first, so factorization is clear there as well.
However, I achieved this with the help of the following hint,
$\textit{Hint: if $r$ in $F$ is a root of such a polynomial, why is $r^2$ also a root?}$
I don't really know why this is the case, any hints as to why this is the case would be appreciated.
Quite simple: $r$ is a root, by definition, if and only if $\;r^3=r+1$.
This implies that $$(r^2)^3=(r^3)^2=(r+1)^2=r^2+1^2=r^2+1\qquad(\text{we're in characteristic }2).$$