Consider the following formulation of the AM-GM Inequalities:
$$f_n(a_1, a_2, a_3,\cdots,a_n) := \left(\sum_{k=1}^n a_k^n\right) - n \prod_{k=1}^n a_k \geqslant 0$$
As required by the classical formulation, we must have $a_k \geqslant 0$
For $n=2,3$, the inequalities admit proof by factorisation
$f_2 \equiv (a_1 - a_2)^2$
$f_3 \equiv (a_1 + a_2 + a_3)(a_1 + a_2 e^{\frac{2 i \pi}{3}} + a_3 e^{\frac{4 i \pi}{3}})(a_1 + a_2 e^{\frac{-2 i \pi}{3}} + a_3 e^{\frac{-4 i \pi}{3}})$
Do any factorisation proofs exist for higher values of $n$?
More precisely, does there exist a complex/real valued, non-trivial factorisation of the homogeneous symmetric polynomials $f_n$, for higher values of $n$, such that the inequality reduces to $\prod |z_j|^2 \geqslant 0$?*
*$z_j$ are appropriate complex/real functions of $a_k$