On the index of an algebraic integer and beyond

Abstract

Let K=Q(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K having minimal polynomial f(x) over Q. For a prime number p, let ip(f) denote the highest power of p dividing the index [AK:Z[θ]]. Let f¯(x)=ϕ¯1(x)e1⋯ϕ¯r(x)er be the factorization of f(x) modulo p into a product of powers of distinct irreducible polynomials over Z/pZ with ϕi(x)∈Z[x] monic. Let the integer l≥1 and the polynomial N(x)∈Z[x] be defined by f(x)=∏i=1rϕi(x)ei+plN(x),N‾(x)≠0¯. In this paper, we prove that ip(f)≥∑i=1ruideg⁡ϕi(x), where ui is a constant defined only in terms of l,ei and the highest power of the polynomial ϕ¯i(x) dividing N‾(x). Further 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 Z by any Dedekind domain.