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

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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 $\Q_p$

if and only if:

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

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

Proof

Necessary Condition

$\Box$

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 $\Q_p$

$\blacksquare$