It is well known that the cyclotomic polynomials encode all the cyclic groups of order $p-1$ for $p$ prime; that is, in the form of a Galois Group of the polynomial $x^p-1$. Concretely we have that
$$F_p(x)=x^p-1 = \prod_{d|p} \Phi_d(x)=(1-x)\left(1+ \sum_{k=1}^{p-1} x^k \right). $$
See "Contemporary Abstract Algebra, Seventh Edition, Joseph A. Gallian, 'Cyclotomic Polynomials, page 562 and Theorem 33.1 on the same page'"
While these are not the only polynomials having the cyclic group of order $p-1$, say $C_{p-1}$, as their Galois Group, this polynomial is in some sense the "simplest" polynomial that encodes $C_{p-1}$ in the form of a Galois Group.
The question: what is the analogue for the alternating group $A_n$? That is, what does $F_q(x)$ look like such that it encodes all the groups of the form $A_q$ in this "simple" or canonical sense? Where $q$ would be the analogue of prime numbers.