Power of Root of Unity Equals Power of Remainder

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 $k \in \Z$.

Then:

$\alpha^k = \alpha^r$

where $0 \le r < n$ is the remainder of $k$ on division by $n$.

Proof

From Division Theorem:

$\exists r, c \in \Z : 0 \le r < n : k = r + c n$

By Definition of Congruence Modulo Integer:

$k \equiv r \pmod n$

From Congruent Powers of Root of Unity are Equal

$\alpha^k = \alpha^r$

$\blacksquare$