AMM Problem 12551 Solution
Number Theory: On the orders of $mod p$ roots of integer polynomials
In this article, I provide the solution to AMM Problem 12551. This article is being
published a day after the AMM deadline!
This is part of my December 31st 2025 deadline, of which I solved 5/7 problems that
I'm sharing here! This is the 5th and final such article.
This problem is a number theory based problem about the orders of roots of polynomials $\mod p$.
We recall the definition for $ord_p(r) = $ the minimum power $k \ge 0$ of $r$ such that $r^k \equiv 1 \mod p$.
For any polynomial $f \in \mathbb{Z}[x]$ with $f(0) \ne 0$, we define
\[ S_f = \left\{ ord_p(r): f(r) \equiv 0 \mod p, p \text{ is a prime}, r \in (\mathbb{Z}/p\mathbb{Z})^{\times} \right\} \]
The problem asks us to prove that $S_f$ is finite iff $f | (x^d-1)^c$ for some positive integers $c,d$.The problem is deeply rooted in both number theory and algebra, requiring us to recall polynomial division, as well as properties of congruence.
Overall, quite interesting!
I hope you enjoyed reading the article
Any feedback is greatly appreciated!