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
This theorem requires a proof. In particular: This is (probably) a specialisation of a more general result on cyclic groups. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44 \beta$