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The polynomial $(x^p-1)/(x-1)$ is irreducible over the integers exactly when $p$ is a prime. But it is only sometimes irreducible when considered as a polynomial over $Z_2$. Are there any simple criteria for $p$ to tell when that happens? What about for other prime moduli besides 2?

user2183078
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    Seems that for $p>2$ , the polynomial is irreducible over $\mathbb Z_2$ if and only if the order of $2$ modulo $p$ is $p-1$ (in other words if $2$ is a primitive root mod $p$). Can someone approve this ? A similar criterion should hold for other moduli. – Peter Oct 15 '22 at 13:38
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    @Peter: yes, that's correct, and the generalization to cyclotomic polynomials over a finite field holds. – Qiaochu Yuan Oct 15 '22 at 17:02
  • I want to add the following. Artin's conjecture predicts that irreducibility holds for a certain percentage of primes. IIRC it is unknown whether the polynomial remains irreducible over $\Bbb{Z}_2$ for infinitely primes $p$, but it IS KNOWN that for infinitely many primes $p$ the polynomial remains irreducible over at least one of $\Bbb{Z}_2$, $\Bbb{Z}_3$, $\Bbb{Z}_5$. – Jyrki Lahtonen Oct 18 '22 at 04:46
  • D'oh, my recollection is confirmed by scrolling down that WP-page :-) – Jyrki Lahtonen Oct 18 '22 at 05:24

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