Ring theory MOC

Group of roots of unity

Let 𝑅 be a commutative ring. The roots of unity

𝜇={𝜁∈𝑅:(∃𝑚∈ℕ)[𝜁𝑚=1]}

form a subgroup of the group of units. ring

Proof

Suppose 𝜁,𝜉 ∈𝜇, then these are both units and (𝜁𝜉)𝑚1𝑚2 =1 where 𝜁𝑚1 =𝜉𝑚2 =1.

In particular, if 𝑅 is the 𝑛th cyclotomic field, we denote this group by 𝜇𝑛.

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