Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$ (https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial).

Now is there a way to characterize for which $(c_0,...,c_n)\in \mathbb F^{n+1} \setminus \{0\}$, is the polynomial $f(X_1,...,X_n)=\sum_{j=0}^n c_je_j \in \mathbb F[X_1,...,X_n]$ reducible in $\mathbb F [X_1,...,X_n]$ ?

For example, if $n$-many among $c_0,...,c_n$ are zero, i.e. if we have $f=e_k$ for some $k$, then we must have $k=n$, because $e_1,...,e_{n-1}$ are all irreducible as seen here Is an elementary symmetric polynomial an irreducible element in the polynomial ring? . Apart from this, I don't know ...