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:
 * $a^n\equiv 1\,(mod\,m)$

Solving the following equation over the smallest integers modulo $m$:
 * $(a^n-1)=(a-1)(\sum^{n-1}_{k=0}a^k)\equiv 0\,(mod\,m)$

will produce the roots.

Table of Roots of Unity Modulo $m$

 * $\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}$

Table of Non-trivial Cubic Roots of Unity Modulo $m$

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

m & a_3 \\ \hline 7 & 2 \\ 13 & 3 \\ 19 & 7 \\ 21 & 4 \\ 31 & 5 \\ 37 & 10 \\ 39 & 16 \\ 43 & 6 \\ 49 & 18 \\ 57 & 7 \\ 61 & 13 \\ 67 & 29 \\ 73 & 8 \\ 79 & 23 \\ 91 & 9 \\ 93 & 25 \\ 97 & 35 \\ \hline \end{array}$

Table of Non-trivial Quartic Roots of Unity Modulo $m$

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

m & a_4 \\ \hline 5 & 2 \\ 8 & 3 \\ 10 & 3 \\ 12 & 5 \\ 13 & 5 \\ 15 & 2 \\ 16 & 7 \\ 17 & 4 \\ 20 & 3 \\ 24 & 11 \\ 25 & 7 \\ 26 & 5 \\ 28 & 13 \\ 29 & 12 \\ 30 & 17 \\ 32 & 15 \\ 34 & 13 \\ \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

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