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I have two questions about solving polynomial equations in a ring $\mathbb{Z_n}$:

1) How to find the number of the roots, without solving the equation? E.g. $x^4-1=0$ (mod 1309) 2) How to solve equations for big n, where simply checking all elements of the ring is not possible?

Moisej Braver
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  • $1309 = 7 \times 11 \times 17$ – Kenny Lau Oct 01 '17 at 08:45
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    What Kenny Lau said. Chinese Remainder Theorem is your friend. Hint/check: $x^4-1$ has two solutions modulo $7$, two modulo $11$, and four modulo $17$. You are expected to CRT-combine them. – Jyrki Lahtonen Oct 01 '17 at 08:46
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    If you don't know the factorization of $n$, then you may be in trouble. If $n$ has a large prime factor $p$, then there are methods for solving this in $\Bbb{Z}_p$. See for example here. But, for a simple polynomial like $x^4-1$ I would probably just pick a random integer $a$ and calculate $a^{(p-1)/4}$ modulo $p$. With a bit of luck we get a primitive fourth root of unity modulo $p$, and we are done. This assuming $p\equiv1\pmod4$. If $p\equiv3\pmod4$ we are just solving $x^2-1\equiv0$ which is trivial. – Jyrki Lahtonen Oct 01 '17 at 08:51
  • And if your modulus is not square-free, then you need to learn about Hensel lifting a solution of congruence modulo $p$ to a solutions of a congruence modulo $p^t$. The process doesn't always work (if there are zeros with multiplicity $>1$), but it is the go-to tool. – Jyrki Lahtonen Oct 01 '17 at 09:10
  • Thanks for the inputs! – Moisej Braver Oct 01 '17 at 09:24

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