Congruent Powers of Root of Unity are Equal
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This article is complete as far as it goes, but it could do with expansion. In particular: There is a group-theoretical result which can be applied directly which is worth adding as a second proof. This ties branches together. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \alpha^l\) |
$\blacksquare$