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

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

Let $F$ be a field.

Let $\alpha \ne 1$ be an $n$-th root of unity.

Then:
 * there exists $k \le n$ such that $\alpha$ is a primitive $k$-th root of unity

Proof
Let $k = \min \set{m : 0 \le m \le n, \alpha^m = 1}$.

$k = 1$.

We have

This contradicts the premise:
 * $\alpha \ne 1$

So:
 * $k \ne 1$

By
 * $\mathop \forall 0 < m < k: \alpha^m \ne 1$

By :
 * $\alpha$ is a primitive $k$-th root of unity