TITLE: When is a linear combination of the elementary symmetric polynomials reducible?
QUESTION [5 upvotes]: 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 ...

REPLY [4 votes]: Suppose that $f=p_1 p_2\cdots p_k$, where each $p_i$ is
irreducible. Since $f$ is a symmetric function, for every $w\in S_n$
and every $i$ we must have $w\cdot p_i = cp_j$ for some $j$, where $c$
is a nonzero constant (depending on $w$ and $i$). In each factor $p_i$
pick out the term of highest degree $d$ that is first in lex order,
e.g., for $d=3$ the lex order is
  $$ X_1^3<X_1^2X_2<X_1^2 X_3<\cdots<X_1^2 X_n<X_1
     X_2^2<\cdots<X_n^3. $$
Unless $p_i$ has the form $a+bX_k$ and no other factor is of this form
(for the same $k$), the highest term (in lex order) of $f$
will have an exponent greater then one, so $f$ is not a linear
combination of $e_j$'s. Thus we must have
  $$ f=c(a+X_1)(a+X_2)\cdots (a+X_n). $$