A generalization of a theorem of Ore

Abstract

Let (K,v) be a discrete rank one valued field with valuation ring R_v. Let L/K be a finite extension such that the integral closure S of R_v in L is a finitely generated R_v-module. Under a certain condition of v-regularity, we obtain some results regarding the explicit computation of R_v-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.