User:Leigh.Samphier/P-adicNumbers/Root of Unity is Primitive Root for Smaller Power

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Theorem

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $F$ be a field.

Let $\alpha \ne 1$ be an $n$-th root of unity.

Then:

there exists $k \le n$ such that $\alpha$ is a primitive $k$-th root of unity

Proof

Let $k = \min \set{m : 0 \le m \le n, \alpha^m = 1}$.

Aiming for a contradiction, suppose $k = 1$.

We have

\(\ds 1\)

\(=\)

\(\ds \alpha^k\)

Definition of Min Operation

\(\ds \)

\(=\)

\(\ds \alpha^1\)

By hypothesis

\(\ds \)

\(=\)

\(\ds \alpha\)

Definition of Power of Field Element

This contradicts the premise:

$\alpha \ne 1$

So:

$k \ne 1$

By Definition of Min Operation

$\mathop \forall 0 < m < k: \alpha^m \ne 1$

By Definition of Primitive Root of Unity:

$\alpha$ is a primitive $k$-th root of unity

$\blacksquare$