Find all roots of the polynomial $f(x)=x^3-[1]_p$ in $\mathbb{Z}/1729\mathbb{Z}$

Question

Problem is the same as in the title, Find roots of the polynomial $f(x)=x^3-[1]_p$ in $\mathbb{Z}/1729\mathbb{Z}$. I am specifically asking only for someone to point me in the direction of the method required to solve this equation and find all of its values, and whether there is an algorithm to determine all 27 of them.

Answer 1

$1729=7\cdot 13\cdot 19$, so solve $x^3=1$ modulo $7$, $13$, and $19$, and combine the solutions using the Chinese Remainder Theorem.

(Obviously both $-12$ and $1$ are solutions, so modulo $7$ and $19$ this gives you two different roots and you can find the third by polynomial division. This won't work modulo 13, but there are not that many possibilities to check by brute force).