Elements of the field $F_2[x] / (x^3 + x + 1)$

Question

What do elements of the field $F_2[x] / (x^3 + x + 1)$ look like? I know this is isomorphic to $F_8$, and that its elements have max degree of 2, so that leaves me with $0$, $1$, $x$, $x^2$, $x+1$ , $x^2+x$, $x^2+1$, $x^2+x+1$. Therefore, is this the same as the field $F_2[x] / (x^3 + x^2 + 1)$?

Answer 1

Hint: Upto isomorphism there is unique field of order $p^n$ for any prime $p$ and any natural number $n$. This simply follows from the fact that any such field is splitting field of the polynomial $x^q-x$ where $q=p^n$