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

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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 Definition of Power of Field Element:

$0^n = 0$

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:

\(\ds \paren{x y^{-1} }^n\) \(=\) \(\ds x^n \paren{y^{-1} }^n\) Common Index Law for Field
\(\ds \) \(=\) \(\ds 1 \times \paren{y^{-1} }^n\) Definition of $n$-th Root of Unity
\(\ds \) \(=\) \(\ds \paren{y^{-1} }^n\) Definition of Unity of Field
\(\ds \) \(=\) \(\ds \paren{y^n }^{-1}\) Negative Index Law for Field
\(\ds \) \(=\) \(\ds 1^{-1}\) Definition of $n$-th Root of Unity
\(\ds \) \(=\) \(\ds 1\)

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 $\struct{F^*, \times \restriction_{F^*}}$

$\blacksquare$