Let me describe my question through an example. Finite field of order (for example) 8 can be constructed as $\mathbb{F}_8 = \mathbb{F}_2[t]/(t^3 + t + 1)$. So one of a natural presentation of the field is the following. $$ \mathbb{F}_8 = \{0, 1, t, t + 1, t^2, t^2 + 1, t^2 + t, t^2 + t + 1\} $$ On the other hand, since the multiplicative group of the field $\langle \zeta \rangle$ is cyclic, the following is also a presentation of the field (as GAP do). $$ \mathbb{F}_8 = \{0, 1, \zeta, \zeta^2, \zeta^3, \zeta^4, \zeta^5, \zeta^6 \} $$ Is there a handy way to get the later presentations from the former ones?
Of course, I can rewrite all the later presentations by relation $t^3 = t + 1$ in advance. But what I would like to do is calculating the later presentation directly when I needed (e.g. when I get the former presentation from matrix multiplication). I would appreciated it if you show me an algorithm for this.
Thank you.
Edit: I rewrite $t$ in the second presentation to $\zeta$. I would also appreciated if you show me a way to find out a primitive $\zeta$ in terms of $t$ (though as DonAntonio already commented, perhaps there is no hope to do so in general).
You can directly check that, talking of the multiplcative group of the field $\;\Bbb F_8\;$ ,we have that $\;ord(t+1)=7\;$ , and thus you can define
$$w:=t+1\implies \Bbb F_8=\{0,1,w,w^2,w^3,w^4,w^4,w^6\}$$
Again, in your post: the first $\;t\;$ (4th line) and the second $\;t\;$ (7-th line) are not necessarily the same, though in this case, passing from one representation to the other, works.