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Say we have roots of unity $\zeta$ and $\rho$ where $o(\zeta)=a$ and $o(\rho)=b$. If $gcd(a,b)=1$, how can we prove that $o(\zeta\rho)=ab$?

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up vote 4 down vote accepted

Suppose $0\lt s\lt ab$. We'll prove $(\zeta\rho)^s\ne1$.

Since $\gcd(a,b)=1$, we know the least common multiple of $a$ and $b$ is $ab$, so $s$ is not a common multiple of $a$ and $b$. Assume, without loss of generality, that it's not a multiple of $a$. Then $bs$ is also not a multiple of $a$, and $(\zeta\rho)^{bs}=\zeta^{bs}\rho^{bs}=\zeta^{bs}\ne1$. So, $(\zeta\rho)^s\ne1$.

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