Let $R=F_2+uF_2+u^2F_2$, where $u^3=0$, be a finite commutative ring. So $R=\{0,1,u,v,uv,u^2,v^2,v^3\}$, where $v=1+u$, $v^2=1+u^2$, $v^3=1+u+u^2$, $uv=u+u^2$. It is well known that
$$x^7-1=(x+v^3)(x^3+uvx^2+v^2x+v^3)(x^3+vx^2+ux+v^3),$$
where all factors in decomposition are basic irreducible polynomials over $R$. Since $x^3+uvx^2+v^2x+v^3$ is a basic primitive polynomial over $R,$ the root $\alpha$ of this polynomial is a primitive element of the extension $R[x]/(x^3+uvx^2+v^2x+v^3).$
In polynomials over fileds, we can find all roots by cyclotomic cosets. But I do not know How can calculate all roots of $x^3+uvx^2+v^2x+v^3$ in ring $R$? Are roots of polynomial $x^3+vx^2+ux+v^3$ as some powers of $\alpha$? how can i calculate them?
Any hints will be appreciated.