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Prove that for all integers $n$, $\cos (2\pi/n)$ is an algebraic number.

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The comment by CropDuster here: mathhelpforum.com/advanced-algebra/… may be of interest. –  01000100 Dec 13 '12 at 10:07
    
@DanielPietrobon: Thank you. –  Metta World Peace Dec 13 '12 at 10:16
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You may look up en.wikipedia.org/wiki/Chebyshev_polynomials. –  Joel Cohen Dec 13 '12 at 10:27
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If you believe that sums and scalar multiples of numbers algebraic over $\mathbb{Q}$ are algebraic, then we have $1 + x + \ldots + x^{n-1}$ has roots $\cos(2\pi/n) \pm i \sin(2\pi /n)$, hence half their sum.

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We have $\zeta_n = e^{2\pi i/n}$ being an algebraic integer for all $n$ simply because it is a root of $x^n -1$. Now notice that $\zeta_n + \zeta_n^{-1} =2\cos(\frac{2\pi i}{n})$ and hence $\frac{1}{2}(\zeta_n + \zeta_n^{-1})$ is contained in the finite algebraic extension $\Bbb{Q}(\zeta_n)$ from which it follows that it itself is algebraic over $\Bbb{Q}$. This comes from the equivalence of the following facts:

Let $E/F$ be a field extension and $\alpha \in E$. TFAE:

  1. $\alpha$ is algebraic over $F$
  2. $\dim_F F[\alpha] < \infty$
  3. $F[\alpha]$ is a field
  4. $F[\alpha] = F(\alpha)$.

Lastly I should add that $\frac{1}{2}(\zeta_n + \zeta_n^{-1})$ is in fact an algebraic integer, meaning to say that its minimal polynomial has coefficients in $\Bbb{Z}$.

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