Construction of the integral closure of a finite integral domain. II

Abstract:

In a previous paper the problem of constructing the integral closure of a finite integral domain $k[{x_1}, \ldots ,{x_n}] = k[x]$ was considered. A reduction to the case $dtk(x)/k = 1,k(x)/k$ separable, and $n = 2$ was made. A subsidiary problem was: if $k[x]$ is not integrally closed, to find a $y$ in $k(x)$ integral over $k[x]$ but not in it. This was done for $n = 2$, but should have been done for arbitrary $n$. The extra details are here given. For the convenience of the reader, the full argument is sketched.