Take the irreducible polynomial $x^3 + x^2 + 1$ over $F_2$ (field of order $2$). Find the splitting field and its roots in that field.
Where I am:
I understand what splitting fields are, and I also know that the splitting field is $F_8$ because $8=2^3$ and we can work in the field $F_2[\theta]/(\theta^3 + \theta^2 + 1)$ in which the polynomial $x^3 + x^2 + 1$ will have linear factors.
What I want to know is: is there are a quick method for finding such roots? Should I simply plug in each element of the splitting field every time given such a question in order to find a linear factor, and work from there? However, I understand that this only works for polynomials up to deg $3$.