$\mathbb{F} = \mathbb{Z_2[x]/(x^3+x+1)}$ is a field.
I need to find an element $a \in \mathbb{F}$ of order $p^n-1$
I know that $\mathbb{F}$ has order $2^3 = 8$ so $a$ must have order 7
ie, $a^7 = 1$
However, I have been mindlessly trying polynomials to see if they have order 7, which as you can imagine lead to scome scary calculations.
Am a missing an important theorem?
First, the elements of $\;\Bbb F\;$ are not polynomials but, if you want, residue classes of polynomials modulo the ideal $\;I:=\langle x^3+x+1\rangle\;$ .
Second, if we denote $\;w:=x+I\in\Bbb F\;$ , then in this field we have the rule $\;w^3=w+1\;$ (why?), so we can write down the elements of the field:
$$\Bbb F=\{0,\;1,\;w\;,w+1\;,w^2\;,w^2+w\;,w^2+w+1,\;w^2+1\}$$
with arithmetic modulo $\;2\;$ and the above rule $\;w^3=w+1\;$. So, for example, we have:
$$\begin{align*}&w\\&w^2\\&w^3=w+1\\&w^4=w\cdot w^3=w(w+1)=w^2+w\;,\;\;etc.\\\end{align*}$$
Of course, in this case there's no problem: any element in $\;\Bbb F^*\;$ which is different of $\;1\;$ is a generator of the field and of the multiplicative group, since the latter is of order $\;7\;$ , a prime (why?) .