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Let us restrict ourselves to $n = p^k$, for some prime $p$. I ask:

  1. what primitive polynomials $P$ vanish identically on $\mathbb Z /n$ ?

  2. what primitive polynomials $P$ vanish identically on $(\mathbb Z /n)^\times$?

Regarding no 2, it should be clear that $P$ could be reduced modulo $X^{\varphi(n)} - 1$, so a possible preliminary question would be: do there exist primitive polynomials of degree less than $\varphi(n)$ that vanish identically on $(\mathbb Z /n)^\times$?

Eric Wofsey
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    Related: https://math.stackexchange.com/questions/3387540/polynomials-that-induce-the-zero-function-mod-n – lhf Oct 25 '21 at 16:52

1 Answers1

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Preliminary question: $x^2 - 1$ vanishes on $(\mathbb Z/8)^\times$ but $\varphi(8) = 4$.

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user985051
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  • As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Oct 26 '21 at 15:02
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    To be clear: $x^2-1$ vanishes on $(\mathbb Z/8)^\times$ (which I assume is the notation here for the residues relatively prime to $8$, but not on all of $\mathbb Z/8$, for example not when $x=2$. – Misha Lavrov Oct 26 '21 at 15:09