User:Leigh.Samphier/P-adicNumbers/Power of Primitive Root of Unity is Primitive Root of Unity for Divisor

Theorem
Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $F$ be a field.

Let $\alpha$ be an primitive $n$-th root of unity.

Let $\beta = \alpha^k$ be a power of $\alpha$ for some $k > 0$.

Let $m = \dfrac n {\gcd \tuple{k, n}}$, where $\gcd \tuple{k, n}$ is the greatest common divisor of $k$ and $n$.

Then:
 * $\beta$ is a primitive $m$-th root of unity

Proof
By :
 * $\exists c \in \N : k = c \cdot \gcd \tuple{k, n}$

We have:

Hence:

By :
 * $\beta$ is an $m$-th root of unity

Let $l \in N : 0 < l < m$.