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I found there that a polynomial in $F[x]$ with $|F| =q $ with degree $n$ will have its roots in $K$ of order $q^n$

Here, I found that either all the roots are primitive or none of them are.

I am looking for a way to build efficiently a polynomial $P_n$ of degree $n$ which roots are of multiplicative order $p^n -1$ in $K$.

If such algorithm exists, is there a way to find among all possible $P_n$, the one(s) with the smallest hamming weight ? (fewest non-zero coefficients)

Any help is welcome

Cyrius Nugier
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    I don't know for sure, but I believe this is a difficult problem in general. A lot of work has gone towards studying primitive trinomials over $\Bbb{F}_2$, but this has never been my specialty, so I cannot say much at all. IIRC a few cases with $n<30$ are known, where no primitive trinomials exist. May be it is even known that those fizzle out? A larger $p$ gives more elbow room. – Jyrki Lahtonen May 14 '20 at 20:52
  • Search 1, 2. – Jyrki Lahtonen May 14 '20 at 20:55
  • Thank you @JyrkiLahtonen, that helped a lot ! – Cyrius Nugier May 15 '20 at 06:36
  • The first post does not say the order of roots have order $q^n$. The multiplicative order of roots of degree $n$ polynomial cannot exceed $q^n -1$. – Sungjin Kim Apr 20 '23 at 19:04

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