On the index theorem of Ore

Abstract

Let K= Q(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and F(x) be the minimal polynomial of θ over the field Q of rational numbers. For a rational prime p, let F(x)≡ϕ1(x)e1…ϕr(x)er(modp) be its factorization into a product of powers of distinct irreducible polynomials modulo p with ϕi(x) ∈ Z[ x] monic. Let ip(F) denote the highest power of p dividing [ AK: Z[ θ] ] and iϕj denote the ϕj-index of F defined by iϕj(F)=(degϕj)Nj, where Nj is the number of points with integral entries lying on or below the ϕj-Newton polygon of F away from the axes as well as from the vertical line passing through the last vertex of this polygon. The Theorem of Index of Ore states that ip(F)≥∑j=1riϕj(F) and equality holds if F(x) satisfies a certain condition called p-regularity. In this paper, we extend the above theorem to irreducible polynomials with coefficients from valued fields of arbitrary rank and give a necessary and sufficient condition so that equality holds in the analogous inequality thereby generalizing similar results for discrete valued fields obtained in Montes and Nart (J Algebra 146:318–334, 1992) and Khanduja and Kumar (J Pure Appl Algebra 218:1206–1218, 2014). The introduction of the notion of ϕj-index of F in the general case involves some new results which are of independent interest as well. © 2016, Springer-Verlag Berlin Heidelberg.