Background
I'm a recreational mathematician whose formal math education is only to pre-university level. I'm very much a self-learner and I particularly like (elementary) number theory.
I was investigating a problem originally asked on Twitter, which was
For each $k$, is there always an $N$ for which the integers less than and coprime to $N$ sum to $kN$? $(k,N\in\mathbb{N})$
The answer to this is no, and it turns out that if Euler's totient function, $\phi(n)$, never has value $2k$, then there is no such $N$ for $k$. The missing values for $k$ are given in http://oeis.org/A079695.
In the comments to sequence A079695 it's stated that "Because the degree of the minimal polynomial of $\cos(2\pi/k)$ is $\phi(k)/2$, the degree can never be a number in this sequence."
I'm familiar with cosine and totient, of course, but not familiar with minimal polynomials. I did a little research on minimal polynomials, but I don't at all understand why a trigonometric function should have any connection to a function, $\phi(n)$, that is essentially about divisibility of positive integers. So...
Question
Is there a straightforward way to explain why these two functions (cosine and totient) are connected via minimal polynomials?