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I have a few questions concerning $\cos(\frac{\pi}{n})$. Are there characterizations for the values $n \in \mathbb{N}$, such that $\cos(\frac{\pi}{n})$

  1. ... is an algebraic number?
  2. ... can be written in terms of square roots?
  3. ... is of the form $a+b \sqrt{d}$ for $a,b \in \mathbb{Q}, d \in \mathbb{N}$?
Martin
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    It is always an algebraic number since $\cos(n\theta)$ is a polynomial (with integer coefficients) in $\cos\theta$. – Dylan Aug 02 '15 at 16:57
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    See https://en.wikipedia.org/wiki/Constructible_polygon#General_theory – lab bhattacharjee Aug 02 '15 at 16:58
  • Thanks so far. So the answer for question 2 is "if and only if the regular $n$-gon is constructible". If further $\varphi(n)=2$, then $n$ also satisfies the condition in question 3 – Martin Aug 04 '15 at 08:59

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Hint: Using $e^{i \theta} = \cos \theta + i \sin \theta$, you can compute $e^{in \theta}$ by taking the $n$th power of the equation for $e^{i \theta}$, and then get you $\cos {n \theta}$ in terms of $\cos \theta$ and $\sin \theta$ by taking the real part of the expression. Then the only detail left is dealing with the fact that $\sin \theta = \sqrt{1 - \cos^2 \theta}$.

user2566092
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