User:Leigh.Samphier/P-adicNumbers/Group of All 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 $U$ denote the set of all roots of unity.

That is, $U = \set{x \in F : \exists n \in \Z_{>0} : x^n = 1}$

Then:
 * $\struct{U, \times \restriction_U}$ is a 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 $x, y \in U$.

By :
 * $\exists n, m \in \Z_{>0} : x^n = y^m = 1$

We have:

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