F J MacWilliams and I deal with this in our book on Error-Correcting Codes See pages 110ff and 124 for example. There are papers by Neal Zierler in the journal "information and Control", for example Zierler, Neal, and John Brillhart. "On primitive trinomials (mod 2)." *Information and Control* 13.6 (1968): 541-554. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Jan 18, 2021 at 7:23 AM Joerg Arndt <arndt@jjj.de> wrote:
Possibly making a fool of myself... Have you checked 'the Bible' (Lidl, Niederreiter)? I seem to recall that there is a nontrivial amount of coverage about binomial polynomials, specifically regarding irreducibility.
Best regards, jj
Let $n = 2^{e_0} q_1^{e_1} \cdots q_i^{e_i}$ be the prime factorization of $n$. Let $p$ be an odd prime. Is it true that $\exists k$ such that $x^n+k$ is irreducible over GF($p$) $\iff$ $(2^{min(e_0,2)} q_1 \cdots q_i) | (p-1)$ ?
Are there similar criteria for "larger" polynomials such as $x^n + k_1 x
* Mike Speciner <ms@alum.mit.edu> [Jan 14. 2021 17:11]: +
k_0$ ?
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