## Compute powers in a field extension

Show that $p(x) = x^3+x+1$ is irreducible over $\mathbb{F}_2$, and let $\theta$ be a root. Compute the powers of $\theta$ in $\mathbb{F}_2(\theta)$.

Note that $p(x)$ has no roots in $\mathbb{F}_2$, and since $p(x)$ has degree 3, it is thus irreducible.

In $\mathbb{F}_2(\theta)$, we have $\theta^3 = \theta+1$. Thus the powers of $\theta$ are as follows.

1. $\theta^1 = \theta$
2. $\theta^2 = \theta^2$
3. $\theta^3 = \theta+1$
4. $\theta^4 = \theta^2+\theta$
5. $\theta^5 = \theta^2 + \theta + 1$
6. $\theta^6 = \theta^2 + 1$
7. $\theta^7 = 1$