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 \\ 4 & 1,-1,a_4,-a_4 \\ \hline \end{array}$

Table of Non-trivial Cubic Roots of Unity Modulo $m$
Solve for $1<a<m$
 * $(a^3-1)=(a-1)(a^2+a+1)\equiv 0\,(mod\,m)$


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

m & a_3 \\ \hline 7 & 2,4 \\ 9 & 4,7 \\ 13 & 3,9 \\ 14 & 9,11 \\ 18 & 7,13 \\ 19 & 7,11 \\ 21 & 4,16 \\ 26 & 3,9 \\ 27 & 10,19 \\ 28 & 9,25 \\ 31 & 5,25 \\ 35 & 11,16 \\ 37 & 10,26 \\ 38 & 7,11 \\ 39 & 16,22 \\ 42 & 25,37 \\ 43 & 6,36 \\ 45 & 16,31 \\ 49 & 18,30 \\ 52 & 9,29 \\ 54 & 19,37 \\ 56 & 9,25 \\ 57 & 7,49 \\ 61 & 13,47 \\ 62 & 5,25 \\ 63 & 4,16,22,25,37,43,46,58 \\ 67 & 29,37 \\ 70 & 11,51 \\ 72 & 25,49 \\ 73 & 8,64 \\ 74 & 47,63 \\ 76 & 45,49 \\ 78 & 55,61 \\ 79 & 23,55 \\ 81 & 28,55 \\ 84 & 25,37 \\ 86 & 49,79 \\ 90 & 31,61 \\ 91 & 9,16,22,29,53,74,79,81 \\ 93 & 25,67 \\ 97 & 35,61 \\ 98 & 67,79 \\ 99 & 34,67 \\ \hline \end{array}$

Table of Non-trivial Quartic Roots of Unity Modulo $m$
Solve for $10}$ 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

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