Prove that tan(Pi/n)/tan(kPi/n) is an algebraic integer when gcd(k,n) = 1

Question

This is equivalent to showing that $\left[ \frac{\zeta -1}{\zeta +1} \right]\left[ \frac{{{\zeta }^{k}}+1}{{{\zeta }^{k}}-1} \right]$ is an integer in the cyclotomic field $\text{Q(}\zeta )$ where $\zeta ={{e}^{2\pi i/n}}$ and gcd(k,n) = 1.

Answer 1

Note that you can assume without loss of generality that $k$ is odd. Namely, if $k$ is even, then $n$ must be odd and you can replace $k$ by $k + n$.

Now since $gcd(k,n) = 1$, you can find an $r$ such that $kr \equiv 1 \pmod{n}$. Then you expression is the same as \begin{equation} \left[\frac{\zeta - 1}{\zeta + 1}\right]\left[\frac{\zeta^k + 1}{\zeta^k - 1}\right] = \left[\frac{(\zeta^k)^r - 1}{\zeta^k - 1}\right]\left[\frac{\zeta^k + 1}{\zeta + 1}\right] \end{equation} But each of the terms in the above expression factors and you obtain \begin{equation} ((\zeta^k)^{r-1} + (\zeta^k)^{r-2} + ... + 1)(\zeta^{k-1} - \zeta^{k-2} + ... - \zeta + 1) \end{equation} which is an algebraic integer.