Group of roots of unity

Let $𝑅$ be a commutative ring. The roots of unity

𝜇 = {𝜁 \in 𝑅 : (\exists 𝑚 \in ℕ) [𝜁 𝑚 = 1]}

form a subgroup of the group of units. #m/thm/ring

Proof

Suppose $𝜁, 𝜉 \in 𝜇$ , then these are both units and $(𝜁 𝜉) 𝑚 1 𝑚 2 = 1$ where $𝜁 𝑚 1 = 𝜉 𝑚 2 = 1$ .

In particular, if $𝑅$ is the $𝑛$ th cyclotomic field, we denote this group by $𝜇 𝑛$ .

#state/tidy | #lang/en | #SemBr