Is the polynomial ring of elementary symmetric polynomials involving n variables over a field is integrally closed

Question

Let $\mathbb{F}$ be any field and $x_1,\ldots,x_n$ be algebraically independent over $\mathbb{F}$ and also let $s_i=i$th elementary symmetric polynomial in the $x_i's$, e.g., $s_1=x_1 + \cdots + x_n;\dots$, $s_n=x_1\cdots x_n$. Now consider the ring $L=\mathbb{F}[s_1,\dots,s_n]$ and its quotient field $K=\mathbb{F}(s_1,\dots,s_n)$. Then how do I show that $L$ is integrally closed in $K$ ? Is $L\cong \mathbb{F}[y_1,\ldots,y_n]$ i.e., an UFD ?

I need some help.

Answer 1

$\mathbb{F}[s_1,\ldots,s_n]$ is just a polynomial ring, as $s_1,\ldots,s_n$ are algebraically independent. So anything that's true of a polynomial ring in $n$ variables is true of $\mathbb{F}[s_1,\ldots,s_n]$.