Congruent Powers of Root of Unity are Equal

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

Let $F$ be a field.

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

Let $k ,l \in \Z$ such that $k \equiv l \pmod n$.

Then:
 * $\alpha^k = \alpha^l$

Proof
By :
 * $\exists c \in \Z : k = l + cn$

We have: