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?.
Therefore it is natural to ask for the elementary symmetric polynomials.
Thanks.
Best Answer
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.