Integer Power of Root of Unity is Root of Unity

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:

$\alpha^k$ is an $n$-th root of unity.

Proof

We have:

\(\ds \alpha^n\)

\(=\)

\(\ds 1\)

By Assumption

\(\ds \paren{\alpha^n}^k\)

\(=\)

\(\ds 1^k\)

\(\ds \paren{\alpha^k}^n\)

\(=\)

\(\ds 1\)

Product of Indices Law for Field

$\blacksquare$