ELMO SL 2014 A7

Find all positive integers $n$ with $n \ge 2$ such that the polynomial $$P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n$$ in the $n$ variables $a_1$ , $a_2$ , $\dots$ , $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.

Proposed by Yang Liu