\begin{definition}[Definition:Root of Unity Modulo m]
Let $n \in \Z_{>0}$ be a positive integer.
Let $m \in \Z_{>1}$ be a positive integer greater than one.
Then $a$ is an '''$n^{th}$ root of unity modulo $m$''' {{iff}}:
:$a^n \equiv 1 \pmod m$
Solving the following equation over the smallest integers modulo $m$:
:$\ds \paren {a^n - 1} = \paren {a - 1} \paren {\sum_{k \mathop = 0}^{n - 1} a^k} \equiv 0 \pmod m$
will produce the roots.
Any root found will also have $a + k m$ as a solution, where $k \in \Z$ is any integer.
Euler's Function $\map \phi m$ root of unity modulo $m$ is the set of all positive integers less than $m$.
{{explain|The above statement needs tightening: what has been described "$\map \phi m$ root of unity modulo $m$" is not clearly defined and (while it possibly may be used to define a set by predicate, is does not appear itself actually to be a set. Difficult to tell.}}
\end{definition}