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 * Mike Speciner <ms@alum.mit.edu> [Jan 14. 2021 17:11]:
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 $x^n + k_1 x + k_0$ ?
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