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.