User:Leigh.Samphier/P-adicNumbers/Power of Primitive Root of Unity is Primitive Root of Unity for Divisor
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Theorem
Let $n \in \Z_{> 0}$ be a strictly positive integer.
Let $F$ be a field.
Let $\alpha$ be an primitive $n$-th root of unity.
Let $\beta = \alpha^k$ be a power of $\alpha$ for some $k > 0$.
Let $m = \dfrac n {\gcd \tuple{k, n}}$, where $\gcd \tuple{k, n}$ is the greatest common divisor of $k$ and $n$.
Then:
- $\beta$ is a primitive $m$-th root of unity
Proof
By Definition of Greatest Common Divisor of Integers:
- $\exists c \in \N : k = c \cdot \gcd \tuple{k, n}$
We have:
\(\ds k m\) | \(=\) | \(\ds c \cdot \gcd \tuple{k, n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c \cdot \gcd \tuple{k, n} \cdot \dfrac n {\gcd \tuple{k, n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c n\) |
Hence:
\(\ds \beta^m\) | \(=\) | \(\ds \paren{\alpha^k}^m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha^{km}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha^{cn}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
By Definition of Root of Unity:
- $\beta$ is an $m$-th root of unity
Let $l \in N : 0 < l < m$.
$\blacksquare$