User:Kip/Sandbox

Definition
Let $n\in\Z_{>0}$ be a positive integer.

Let $m\in\Z_{>1}$ be a positive integer greater than one.

Then $a$ is said to be an $n^{th}$ root of unity modulo $m$ if:
 * $a^n\equiv 1\,(mod\,m)$

Table of Roots of Unity

 * $\begin{array}{|c||c|} \hline

n & a \\ \hline 1 & 1 \\ 2 & 1,-1 \\ 3 & 1,$a_3$,$-a_3-1$ \\ 4 & 1,-1,$a_4$,$-a_4$ \\ \hline \end{array}$

Theorem
Let $a\in\Z$ be an $n^{th}$ root of unity modulo $m$

Let $A\in\Z_{>0}$ be a positive integer coprime with $m$

Let $\alpha\in\Z_{>0}$ be a positive integer

Let $n\in\Z_{>0}$ be a positive integer that is a factor of $\alpha\phi(m)$
 * $A^{\frac{\alpha\phi(m)}{n}}\equiv a\,(mod\,m)$

Proof

 * $A^{\alpha\phi(m)}\equiv a^n\,(mod\,m)$