Equivalence of Definitions of Primitive Root of Unity

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

Let $F$ be a field.

Let $U_n$ denote the set of all $n$-th roots of unity.

Definition 1 implies Definition 2
Let $\alpha \in U_n$ such that:
 * $U_n = \set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}$

By :
 * $\alpha = \alpha^1$

Hence:
 * $U_n = \set{1, \alpha^1, \alpha^2, \ldots, \alpha^{n-1}}$

By :
 * $\forall m \in \N: 0 < m < n : \alpha^m \ne 1$

Definition 2 implies Definition 1
Let $\alpha \in U_n$ such that:
 * $\forall m \in \N : 0 < m < n: \alpha^m \ne 1$

$\exists m, k \in \N : 0 \le m < k < n : \alpha^k = \alpha^m$
 * $\alpha^{\paren{k - m}} = \alpha^k \circ \alpha^{-m} = \dfrac {\alpha^k} {\alpha^m} = 1$

$0 < k-m < n$, a contradiction

$\forall m, k \in \N: 0 \le m < k < n : \alpha^k \ne \alpha^m$

$\set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}} \subseteq U_n$

From Cardinality of Superset of Finite Set:
 * $n = \card {\set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}} \le \card {U_n}$

Consider the polynomial $X^n - 1$ over $F$.

From Polynomial over Field has Finitely Many Roots:
 * The polynomial $X^n - 1$ has at most $n$ roots.

So:
 * $\card {U_n} \le n$

Hence:
 * $\card {U_n} = n = \card {\set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}}$

From Tests for Finite Set Equality:
 * $U_n = \set{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}}$