On ramified complete discrete valuation rings

Abstract

R becomes a topological ring by defining the neighborhoods of zero to be the ideal (q) and all powers of (q). R is a complete discrete valuation ring if it is complete with respect to this topology. A non-Archimedean discrete valuation V can be defined on R by letting V(0) = co and V(a) (a^O) be the highest power of q to divide a. This valuation can be extended in the natural way to the quotient field K of R. The nonzero elements of R are then the quantities in K with non-negative valuation. Conversely, given a field K with nonArchimedean discrete valuation, the elements in K with non-negative valuation are the nonzero elements of a discrete valuation ring R, the ring of integers of K. R is complete if and only if K is complete. Again K is the quotient field of R. It is largely in this context that complete discrete valuation rings have been studied. It has been shown that if R is complete and has the same characteristic as its residue class field F then R is isomorphic to the ring of power series F[[x]]. Moreover, in the remaining case (R has characteristic infinity and F has characteristic p),Ris uniquely determined by F if R is unramifiied, i.e., if p is prime in R. Also there exists an R for any given F. If V(p) =n, »> 1, R is said to be ramified with ramification index n. For references see [2]. This paper is concerned with some aspects of the structure of ramified complete discrete valuation rings. Throughout this paper, the symbols R, R', Rn etc., will denote complete discrete valuation rings of characteristic zero having the same residue field F of characteristic p. The subscript on the ring symbol will designate the ramification index or if there is none the ring is unramified. Rings Rn have been studied extensively, largely as a part of algebraic number theory. Thus, as indicated below, a number of the results of this paper are known, however the methods are new and simple. Theorem 1 provides a description of an arbitrary ring Rn in terms of the unique unramified R and is closely related to the theorem [3, p. 237, Theorem 11 ] which states that every Rn is an R(t) where t is a root of an Eisenstein equation. This characterization is then used