User:Leigh.Samphier/P-adicNumbers/Cyclic Group of All n-th Roots of Unity

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

Let $\struct{F,+,\times}$ be a field with zero $0$ and unity $1$.

Let $F^* = F \setminus \set 0$.

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

Then:
 * $\struct{U_n, \times \restriction_{U_n}}$ is a cyclic subgroup of $\struct{F^*, \times \restriction_{F^*}}$

Proof
By :
 * $0^n = 0$

Hence:
 * $0 \notin U_n$

Thus:
 * $U_n \subseteq F^*$

From Multiplicative Group of Field is Abelian Group:
 * $\struct{F^*, \times \restriction_{F^*}}$ is an Abelian group

Let $x, y \in U_n$.

We have:

From One-Step Subgroup Test:
 * $\struct{U_n, \times \restriction_{U_n}}$ is a subgroup of $\struct{F^*, \times \restriction_{F^*}}$

show finite

From Finite Multiplicative Subgroup of Field is Cyclic: $:\struct{U_n, \times \restriction_{U_n}}$ is a cyclic subgroup of $\struct{F^*, \times \restriction_{F^*}}$