Let $k$ be an algebraically closed field of characteristic p. Let $Z\subset k[x_1,\cdots,x_n]$ be a $k$-subalgebra of a polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in $k[x_1,\cdots,x_n]$) is again an element of $Z.$ Then is $Z[x_1^p,\cdots,x_n^p]$ a normal ring? For the easiest case of $Z=k[f]-$where $f$ is an irreducible polynomial, the answer is yes, since $Z[x_1^p,\cdots,x_n^p]$ in this case is a complete intersection and regular in codimension 2.

Let $k$ be an algebraically closed field of characteristic p. Let $Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in $k[x_1,\cdots,x_n]$) is again an element of $Z.$ Then is $Z[x_1^p,\cdots,x_n^p]$ a normal ring? For the easiest case of $Z=k[f]-$where $f$ is an irreducible homogeneous polynomial, the answer is yes, since $Z[x_1^p,\cdots,x_n^p]$ in this case is a complete intersection and regular in codimension 2.