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 $l \in N : 0 < l < m$.

finish

$\blacksquare$