If I’m not mistaken, Conway’s notion of Nim-multiplication gives a way of singling out a “first” degree n irreducible polynomial over GF(2). For instance, x = *4 satisfies x^2 = *6, x^3 = *14, and x^4 = *5, so the first irreducible polynomial of degree 4 is x^4 + x + 1. It would be interesting to peruse these polynomials for the first dozen powers of 2 to see if there’s a simple pattern. When n isn’t a power of 2 you need to use Nim-values associated with infinite ordinals but that’s okay since they’re well-ordered. Jim Propp 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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