Is $K[x_1, x_2,...]$ normal or not? As the title speaks for itself, is the polynomial ring in infinitely many variables over a field normal or not?
Can someone provide a reference/proof? Thanks
Also, what about a directed union of normal subrings? Is that normal?
Tangential to this, what's a criterion for an element in $K[x_{1},x_{2},\ldots]$ to be nilpotent? We know that in $K[x]$ we have that $f$ is nilpotent if and only if the coefficients are nilpotent and similarly in finitely many variables. Does that hold in our case too?
 A: If $R$ is an UFD (unique factorization domain), then $R$ is integrally closed (see here). The polynomial rings in finitely many indeterminates are UFDs. Moreover, $K[X_1,\dots,X_n,\dots]$ is also an UFD (why?), so it is integrally closed.
A direct union of integrally closed rings is integrally closed. 
Remark. If $K$ is a field, as I suppose to be, then $K[X_1,\dots,X_n,\dots]$ is an integral domain and its only nilpotent element is $0$. In general, if $K$ is a commutative ring, then you have the same criterion as for the polynomial rings in finitely many indeterminates.
A: Since the polynomial ring in finitely many variables over a field is a UFD, it is normal.
Hence it suffices to prove that a directed union of normal subrings is normal.
Let $A$ be a directed union of a family of normal subrings $(A_i)_{i \in I}$.
Since every $A_i$ is an integral domain, $A$ is also an integral domain.
Let $K$ be the field of fractions of $A$.
Suppose $\alpha \in K$ is integral over $A$.
There exists $c_1, \cdots, c_n \in A$ such that $\alpha^n + c_1\alpha^{n-1} + \cdots + c_n = 0$.
$\alpha$ can be wriiten as $\alpha = \frac{a}{b}$, where $a, b \in A$.
Since $(A_i)_{i\in I}$ is directed, there exists $i \in I$ such that $a, b, c_1,\dots,c_n \in A_i$.
Hence $\alpha$ is contained in the field of fractions of $A_i$ and it is integral over $A_i$.
Since $A_i$ is normal, $\alpha \in A_i$.
Hence $A$ is normal.
