How to reduce a polynomial over an extension field (with or without MAGMA)

Question

I have the elements $u^{4770}$ and $u^{7489}$ lying in the finite field $\mathbb{F}_{29^3} = \mathbb{F}_{29}[u]$ where $u^3+2u+27=0$. I'd like to find equivalent values with lower degrees so that I can work with them more easily, but I don't know how to do this. I know that in theory, I could do the following:

\begin{align*} u^{4770} &= u^{4767}*(-2u-27) \\ \end{align*} and keep on reducing the highest power of $u$ until I get something reasonable. This seems like a lot of work, and I'm hoping there's a more clever way of doing this.

I do have access to MAGMA programming language, so if there's a way to do it through that, I'd welcome that as a solution as well.

Answer 1

In Magma, the intrinsic Eltseq can be used to return the coefficients of an element of an extension field considered as a vector over its base field. For your example:

> F<u> := ext<GF(29) | Polynomial([27,2,0,1])>;
> x := u^4770;
> x;
u^4770
> Eltseq(x);
[ 26, 10, 12 ]
> x2 := 26 + 10*u + 12*u^2;
> x2;
u^4770
> x2 eq x;
true