User:Leigh.Samphier/P-adicNumbers/Group of All 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 $U$ 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 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 $x, y \in U$.

By Definition of Root of Unity:

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

We have:

\(\ds \paren{x y^{-1} }^{nm}\)

\(=\)

\(\ds x^{nm} \paren{y^{-1} }^{nm}\)

Common Index Law for Field

\(\ds \)

\(=\)

\(\ds 1 \times \paren{y^{-1} }^{nm}\)

Integer Power of Root of Unity is Root of Unity

\(\ds \)

\(=\)

\(\ds \paren{y^{-1} }^{nm}\)

Definition of Unity of Field

\(\ds \)

\(=\)

\(\ds \paren{y^{nm} }^{-1}\)

Negative Index Law for Field

\(\ds \)

\(=\)

\(\ds 1^{-1}\)

Integer Power of Root of Unity is Root of Unity

\(\ds \)

\(=\)

\(\ds 1\)

From One-Step Subgroup Test:

$\struct{U, \times \restriction_U}$ is a subgroup of $\struct{F^*, \times \restriction_{F^*}}$

$\blacksquare$