Why are the coefficients of the cyclotomic polynomial symmetric ?
$\Phi_n(x):=\frac{x^n-1}{\prod\limits_{d |n, d<n}\Phi_d(x)}\ $or $\Phi_n(x)=\frac{x^n-1}{lcm\{x^d-1:d|n,0<d<n\}}$
so we use one of the definitions above, I see some paper, where a different definition is given and the claim is proved by using möbius function, is there another proof without involving möbius or so ?
By definition, $\Phi_n(x) = \frac{x^n - 1}{\prod\limits_{d | n, d < n}\Phi_d(x)}$.
We can rewrite this as $\Phi_n(x) = \frac{x^{n - 1} + \ldots + 1}{\prod\limits_{d | n, 1 < d < n}\Phi_d(x)}$. Note that the numerator is symmetric.
Recall that the product of symmetric polynomials is symmetric, as is the quotient if they divide evenly.
From there, we can use strong induction to show that each cyclotomic polynomial is the quotient of two symmetric polynomials.
Thus, $\Phi_n$ is symmetric.