On the index of an algebraic integer and beyond

Abstract

Let K=Q(θ)be an algebraic number field with θin the ring AKof algebraic integers of Khaving minimal polynomial f(x)over Q. Fo r a prime number p, let ip(f)denote the highest power of pdividing the index [AK:Z[θ]]. Let ̄f(x) = ̄φ1(x)e1··· ̄φr(x)erbe the factorization of f(x) modulo pinto a product of powers of distinct irreducible polynomials over Z/pZwith φi(x) ∈Z[x]monic. Let the integer l≥1and the polynomial N(x) ∈Z[x]be defined by f(x) =r∏i=1φi(x)ei+plN(x), N(x) = ̄0. In this paper, we prove that ip(f) ≥r∑i=1uidegφi(x), where uiis a constant defined only in terms of l, eiand the highest power of the polynomial ̄φi(x)dividing N(x). Fu r t h e r a class of irreducible polynomials is described for which the above inequality becomes equality. The results of the paper quickly yield the well known Dedekind criterion which gives a necessary and sufficient condition for ip(f)to be zero. In fact, these results are proved in a more general set up replacing Zby any Dedekind domain.