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?