User:Leigh.Samphier/P-adicNumbers/Characterization of Primitive m-th Root of Unity in P-adic Numbers

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $m \in \N$ such that $p \nmid m$.

Then:
 * there exists an $m$-th primitive root of unity in $\Q_p$


 * $m \divides \paren{p-1}$
 * $m \divides \paren{p-1}$

In which case, an $m$-th root of unity in $\Q_p$ is a $\paren{p-1}$-th root of unity in $\Q_p$

Sufficient Condition
Let $m \divides \paren{p-1}$.

By definition of divisor:
 * $\exists k \in \N: k \ge 1 : p-1 = km$

Hence:
 * any $m$-th root of unity in $\Q_p$ is a $\paren{p-1}$-th root of unity in $\Q_p$