what i want to know is how to compute the powers of $\theta$ in $F_2$ and also how many powers am i looking to compute. How can i find such powers This a new chapter that we are learning and the teacher is flying through it and this are also new concept that i have just learn and i was wondering if i can have some guidance in this problem.     Show that $x^3 + x + 1$ is irreducible over $F_{2}$ and let $\theta$ be a root. Compute the powers of $\theta$ in $F_{2} (\theta)$.    Proof:  $F_2$ has only two element $0$ and $1$ thus to show that the polynomial is irreducible i did $$0^3+0+1 = 1 \not=0$$ and $$ 1^3+1+1 = 3 = 1 \not=0$$ since there is no root therefore the polynomial is irreducible.   Now my question is how to you go about computing the powers of $\theta$
$x^3+x+1$ irreducible over $\mathbb{F}_2$, $\theta$ root, compute powers of $\theta$ Problem statement:     Show that $x^3+x+1$ is irreducible over $\mathbb{F}_2$ and let $\theta$ be a root. Compute the powers of $\theta$ in $\mathbb{F}_2$.   I am having trouble computing the powers of $\theta$. $x^3+x+1$ is irreducible, since any cubic polynomial over a field is reducible if and only if it has a root in the field. Since $\mathbb{F}_2=\{0,1\}$, we see that none of these is a root of $x^3+x+1$.  To compute the powers of $\theta$, note that $\theta^3+\theta+1=0$. Hence $\theta^3=-(\theta+1)$. Is this a good enough answer for $\theta^3$?  To compute $\theta^2$, note that $\theta = -(\theta^3+1)$. Therefore, $$\theta^2=(\theta^3+1)^2 = \theta^6 + 2\theta^3 + 1 = (-\theta-1)^2-2\theta-2+1,$$ which just reduces to $\theta^2$. Any ideas for computing these powers?  Thanks!
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