Inventiones math. 19, 235 - 249 (1973) , 9 by Springer-Verlag 1973 Eliminating Wild Ramification Helmut P. Epp* (Chicago) Introduction The purpose of this paper is to prove the following theorem: Let S and R be two discrete valuation rings such that (i) S dominates R, and (2) if ~ and R are the residue fields of S and R respectively and if the characteristic p of ~ is not zero, then the largest perfect subfield ~P~ of ~ is separable and algebraic over R. Then there exists a discrete valuation ring Twhich is a finite extension of R such that the localizations of the normalized join of S and T are weakly unramified over T. The restriction on the residue field is quite mild and is satisfied by virtually all field extensions occurring in algebraic geometry and formal algebraic geometry (see 0.4). In case S is tamely ramified over R, i.e., the ramification index of S over R is prime to the residue characteristic, then this theorem is essentially Abhyankar's lemma. Geometric applications of this theorem will be given by the author in a subsequent paper. This theorem arose in attempting to give a direct proof of the existence of stable reduction of curves. The proof given by Mumford and Deligne in [9] is indirect and depends on the (unpublished) proof of the existence of stable reduction of abelian varieties due to Grothendieck. Another proof due to Artin and Winters (see [2]) depends on a numerical analysis of the types of special fibers of curves over discrete valuation rings. A direct proof by a local analysis runs into dif- ficulties caused by wild ramification (see [2"1). The proof of the theorem relies on the exploitation of the classical theory of discrete valuation rings. In Section 0 we review the basic facts used and introduce a new condition on field extensions in characteristic p 4:0. In Section 1 the theorem is proved for the case of complete discrete valuation rings, the proof being split into two parts: equal and unequal characteristic cases (IA and 1B respectively). In the equal characteristic case we use power series and Artin-Schreier theory. In the unequal characteristic case we introduce a novel Hensel expansion of principal units which is useful in the study of principal units when the residue field is not perfect. We also use Kummer theory. In Section 2 we show how to descend to the non-complete case. Finally in Section 3 we give * Supported in part by NSF GP 28420.