Suppose $p(x)$ is an irreducible polynomial over a field $F$. Let $\alpha$ be a root. Compute the powers of $\alpha$ in $F(\alpha)$.
I am not sure what the powers of a root are and how to compute them. I read an example with $p(x)=x^3+x+1$ in $F_2$, and the powers of $\alpha$ are:
$\alpha^0 = 1$
$\alpha^1 = \alpha$
$\alpha^2 = \alpha^2$
$\alpha^3 = \alpha+1 \quad$ (I can guess this is from setting $p(\alpha)=0$ with coefficients in $F_2$ and solve for $\alpha^3$)
$\alpha^4 = \alpha^2+\alpha$
$\alpha^5 = \alpha^2+\alpha+1$
$\alpha^6 = \alpha^2+1$
$\alpha^7 = 1$.
Can someone please show me where these numbers come from?
I am sorry if this question was asked. Thank you.
Simply start with $\alpha^0=1$ and repeatedly compute $\alpha^{n+1}=\alpha\cdot\alpha^n$, where you "mod out" $\alpha^{\deg p}$ using the given polynomial $p$ if it appears.