Let $S=\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ be a polynomial ring. Let $e\_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$.


For $n=2$:


$e\_1=x\_1+x\_2$;  

$e\_2=x\_1x\_2$.


For $n=3$:


$e\_1=x\_1+x\_2+x\_3$;  

$e\_2=x\_1x\_2+x\_1x\_3+x\_2x\_3$,  

$e\_3=x\_1x\_2x\_3.$


In general for any $n$ and $a$, one has
$$ e\_a(x\_1,x\_2,\dots,x\_n):=\sum\_{1 \leq i\_{1} < i\_{2} < \cdots < i\_a \leq n} x\_{i\_1}x\_{i\_2}\cdots x\_{i\_a}
$$


**Question**: Let $n \in \mathbb{N}$ with $n \geq 3$. Is it true that $e\_a$ is an irreducible element in $\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ for $a=2,3,\dots,{n-1}$.


For $n=1$, $e\_1$ is an irreducible element.
For $n=2$, $e\_1$ is an irreducible element.
For $n=3$, $e\_1$ and $e\_2$ are irreducible element. 


Fact: $e\_1$ is be definition, an irreducible element. And, $e\_n$ is trivially reducible. 
My Question is therefore, to know, whether $e\_2,e\_3,\dots,e\_{n-1}$ are irreducible elements in $\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ for $n \geq 4$. 


Similar results: Power sum symmetric polynomials and complete homogeneous symmetric polynomials are irreducible elements in $\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ for $n \geq 3$.
For complete symmetric polynomial, see [Is complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?](https://mathoverflow.net/questions/98043).


Therefore it is natural to ask for the elementary symmetric polynomials.


Thanks.



Is an elementary symmetric polynomial an irreducible element in the polynomial ring?
For $\alpha\neq n$, the symmetric polynomial is of the form $f\cdot x\_n + g$ where $f,g$ are non-zero elements of $A={\mathbb C}[x\_1,...,x\_{n-1}]$ with no common factor.


Thus $${\mathbb C}[x\_1,...,x\_n]/(e\_\alpha)=A[x\_n]/(f x\_n+g)=A[g/f]\subset K$$


where $K$ is the quotient field of $A$. It follows that ${\mathbb C}[x\_1,...,x\_n]/(e\_\alpha)$ is a domain, so $e\_\alpha$ is irreducible. 


