Take an odd prime $p$ and put $x:=\sum\limits_{j=1}^{p-1}\left(a_{1j}\sqrt{p}\cos\dfrac{j\pi}p+a_{2j}\sin\dfrac{j\pi}p +a_{3j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes the minimal polynomial of $x$, can we prove that $p$ divides all coefficients of $f$ except the leading one?

I have quite a bit of numerical evidence for this. Note that it obviously doesn't hold without the $\sqrt p$ factor, but more interestingly, it is also false if $\sqrt p$ goes with the other terms instead of the $\cos$ term. Moreover, it seems that in those cases, none of the coefficients is divisible by $p$.

(More generally, I think those coefficients are divisible by $p$ if we replace $\dfrac{j\pi}p$ by $\dfrac{j\pi}{p^r},\ r\in\mathbb N$ and do the sum over $j=1,...,p^r-1$.)

If all but one of the $a_{ij}$ are $0$, the claim is quite easy to prove (and not new). For instance, for $x=\sin\dfrac{j\pi}p$ with any fixed $j$, we have explicitly $$f(x)=\sum\limits_{i=0}^k(-1)^k\dbinom p{2i+1}(1-x^2)^{k-i}x^{2i},$$ where $p=2k+1$. So the claim is obvious here.

Added: It should be clear from Galois theory that the conjugates of $x$ are the sums obtained by replacing all the $j$'s by $kj$ for a fixed $k=2,...,p-1$.

Literature:

Beslin, S., de Angelis, V., 2004. The minimal polynomials of sin(2π/p) and cos(2π/p). Mathematical Magazine 77, 146–149.

Heierman, William E., Minimal polynomials for trig functions of angles rationally commensurate with π

Lang, Wolfdieter, Minimal Polynomials of sin (2π/n)

Surowski, David, and McCombs, Paul, Homogenous polynomials and the minimal polynomial of cos(2π/n)

W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.