On Dedekind criterion and simple extensions of valuation rings

Abstract

Let R be an integrally closed domain with quotient field K and S be the integral closure of R in a finite extension L = K(θ) of K with θ integral over R. Let f(x) be the minimal polynomial of θ over K and p be a maximal ideal of R. Kummer proved that if S = R[θ], then the number of maximal ideals of S which lie over p, together with their ramification indices and residual degrees can be determined from the irreducible factors of f(x) modulo . In this article, the authors give necessary and sufficient conditions to be satisfied by f(x) which ensure that S = R[θ] when R is the valuation ring of a valued field (K, v) of arbitrary rank. The problem dealt with here is analogous to the one considered by Dedekind in case R is the localization of Z at a rational prime p, which in fact gave rise to Dedekind Criterion (cf. [99. Montes , J. , Nart , E. ( 1992 ). On a theorem of ore . J. Algebra 146 : 318 – 334 . [CrossRef], [Web of Science ®] View all references]). The article also contains a criterion for the integral closure of any valuation ring R in a finite extension of the quotient field of R to be generated over R by a single element, which generalizes a result of Dedekind regarding the index of an algebraic number field.