### 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 $R=\z$.