Dedekind's criterion and Integral bases

Abstract

Let $R$ be a principal ideal domain with quotient field $K$, and
$L=K(\al)$, where $\al$ is a root of a monic irreducible
polynomial $F(x)\in R[x]$. Let $\z_L$ be the integral closure of
$R$ in $L$. In this paper, for every prime $p$ of $R$, we give a
new efficient version of Dedekind's criterion in $R$, i.e.,
necessary and sufficient conditions on $F(x)$ to have $p$ does
not divide the index $[\z_L:R[\al]]$. Some computational examples
are given for $R=\z$.