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