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For $n,k,m\ge 1$ integer, define $$S(n,m,k)=\sum_{A\subset\{1,\ldots,n\},\\\#A=k}\left(\prod_{a\in A}(-1)^a2^m\cos^m\left(\frac{a\pi}n\right)\right).$$ In other words, $S(n,m,k)$ is the $k$'th elementary symmetric polynomial evaluated in $$-2^m\cos^m\left(\frac{\pi}n\right),2^m\cos^m\left(\frac{2\pi}n\right),\ldots,(-1)^n2^m\cos^m\left(\frac{n\cdot n\pi}n\right).$$ It seems like all $S(n,m,k)$ are integer. Can anyone prove or refute this?

If they are all integer, I'd like to find some 'combinatorical' formula for $S(n,m,k)$, containing only polynomials, powers of integers and factorials.

The closest I've gotten is the following identity: $$2+\prod_{j=1}^{2n}\left(X-2\cos\left(\frac{\pi k}n\right)\right)=2\sum_{j=0}^nX^{2n-2j}(-1)^j\frac{n(2n-j-1)!}{(2n-2j)!j!}$$

Mastrem
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    $2\cos(q)$ is an algebraic integer for any rational $q$, hence so is the value of the symmetric polynomial. It is also rational, because the Galois group of whatever extension contains all the cosines permutes them. Hence this is an integer. – Wojowu Dec 21 '18 at 09:54
  • ... meant $2\cos(q\pi)$. – metamorphy Dec 21 '18 at 10:14

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