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I would like to know, if there are any intuitive fast approaches to finding generator elements of small finite extension fields. Like for example, i don't want to try every element of lets say

$\mathbb{F}_p$ and $F_q[x]$ where the coefficients of F are elements of $\mathbb{F}_p$ and $q = p^n$ with $n=2$ and $p=3$. How do i find the generator fast and efficiently for $F_q[x]$(should be obvious, but somehow it is not)? The elements in $F_q[x]$ namely 0 and 1 can not be solutions, but for example $\bigcup_{i=1}^q g^i = F_q[x]$ ... how do i find this g fast and efficiently, without just blindly guessing and trying elements?

This edit will probably be downvoted, but whatever ... this question is kind of easy to understand ... but sure, i'll edit it ...

Gerry Myerson
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Clebo Sevic
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    Math SE does not appreciate people shouting in their questions. If you want to receive better responses, I suggest editing your question body. – Clement Yung Feb 15 '20 at 11:58
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    This is confusing. You don't want to try every element of ${\bf F}_3$, but there are only three elements in ${\bf F}_3$. You don't want to try every element of ${\bf F}_n[x]$, but that's a ring of polynomials, and not a field at all. What do you really mean? Please edit the body of the question to clarify. – Gerry Myerson Feb 15 '20 at 11:58
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    In the context of field and ring extensions there are many kind of generators : generator of cyclic group, of vector space (basis), generator as algebra, generator as field. So what do you want exactly ? An element of $\Bbb{F}{3^2}^\times$ of multiplicative order $3^2-1$ ? If so it is $-(\sqrt{-1}+1)= (\sqrt{2}+\sqrt{-2})/2=e^{2i\pi/8} \bmod 3$. The multiplicative monoid of $\Bbb{F}{3^2}[x]$ is not a group, its units subgroup is $\Bbb{F}_{3^2}^\times$. – reuns Feb 15 '20 at 14:47
  • If you are asking about primitive elements (= a generator of the multiplicative group), then this is not always easy. Small fields can be handled with techniques like these, but bigger fields are more taxing. – Jyrki Lahtonen Feb 15 '20 at 16:54

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