Subgroup of Circle Group Generated by Distinct Roots of Unity

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Theorem

Let $K$ be the circle group.

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $d = \lcm \set {m, n}$ be the least common multiple of $m$ and $n$.


Let $\alpha$ be a primitive $n$th root of unity.

Let $\beta$ be a primitive $m$th root of unity.

Let $\gamma$ be a primitive $d$th root of unity.


Let $H = \gen {\alpha, \beta}$ be the subgroup of $K$ generated by $\alpha, \beta$.


Then $H = \gen \gamma$.


Proof




Sources