Congruent Powers of Root of Unity are Equal

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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 Definition of Congruence Modulo Integer:

$\exists c \in \Z : k = l + c n$

We have:

\(\ds \alpha^k\) \(=\) \(\ds \alpha^{\paren {l + c n} }\)
\(\ds \) \(=\) \(\ds \alpha^l \alpha^{\paren {c n} }\) Sum of Indices Law for Field
\(\ds \) \(=\) \(\ds \alpha^l \cdot 1\) Integer Power of Root of Unity is Root of Unity
\(\ds \) \(=\) \(\ds \alpha^l\)

$\blacksquare$