Dedekind’s Criterion and Integral Bases

Abstract

Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL 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 not dividing the index [ℤL: R[α]], for every prime p of R. Some computational examples are given for R = ℤ.