On the index theorem of Ore

Abstract

Let $$K=\mathbb {Q}(\theta )$$ be an algebraic number field with $$\theta $$ in the ring $$A_K$$ of algebraic integers of K and F(x) be the minimal polynomial of $$\theta $$ over the field $$\mathbb {Q}$$ of rational numbers. For a rational prime p, let $$ F(x)\equiv \phi _1(x)^{e_1}\ldots \phi _r(x)^{e_r}(\mod p)$$ be its factorization into a product of powers of distinct irreducible polynomials modulo p with $$\phi _i(x)\in \mathbb Z[x]$$ monic. Let $$i_p(F)$$ denote the highest power of p dividing $$[A_K:\mathbb {Z}[\theta ]]$$ and $$i_{\phi _j}$$ denote the $$\phi _j$$ -index of F defined by $$i_{\phi _j}(F)= (\deg \phi _j)N_j$$ , where $$N_j$$ is the number of points with integral entries lying on or below the $$\phi _j$$ -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 $$i_p(F)\ge \sum \nolimits _{j=1}^{r}i_{\phi _j}(F)$$ 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 $$\phi _j$$ -index of F in the general case involves some new results which are of independent interest as well.