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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.

dh16
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    See here. I try to serve the community with that Q&A pair. I'm not sure it is clear enough, because the talk about discrete logarithm, while relevant, may obscure the discussion for some readers. This very example is done there. I welcome all comments so that I can improve that thread. – Jyrki Lahtonen Apr 22 '15 at 18:47

1 Answers1

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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.

Hagen von Eitzen
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  • Thank you for your help. I can see most of the terms, but there is a term I am still confused. $\alpha^3 = \alpha \alpha^2$, so what happens if I "mod out" $\alpha^3$, where $3$ is the degree of $p$? I think I can use the polynomial, but it would be nice if I understand it in general. Thanks again. – dh16 Apr 22 '15 at 18:45