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 𝔭. 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. [9]). 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.