Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T00:49:22.281Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial Isomorphisms of V-Rings

Published online by Cambridge University Press:  20 November 2018

N. Heerema
N. Heerema*
Affiliation:
Florida State University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 529 - 539
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Cohen, I. S., Structure of complete local rings, Trans. Amer. Math. Soc., 59 (1946), 54–106.Google Scholar
2. Heerema, N., On ramified complete discrete valuation rings, Proc. Amer. Math. Soc., 10 (1959), 490–496.Google Scholar
3. Heerema, N., Derivations and embeddings of a field in its power series ring, 11, Michigan Math. J., 8 (1961), 129–134.Google Scholar
4. Heerema, N., Derivations on p-adic fields, Trans. Amer. Math. Soc., 102 (1962), 346–351.Google Scholar
5. Heerema, N., Embeddings of a p-adic field and its residue field in their power series rings, Proc. Amer. Math. Soc., 14 (1963), 37–43.Google Scholar
6. Heerema, N., Derivations and inertial automorphisms of complete regular local rings, Amer. J. Math., 88 (1966), 33–42.Google Scholar
7. MacLane, S., Subfields and automorphism groups of p-adic fields, Ann. of Math., 40 (1939), 423–442.Google Scholar
8. Neggers, J., Derivations on p-adic fields, Trans. Amer. Math. Soc., 115 (1965), 496–504.Google Scholar
9. Wishart, E., Higher derivations on p-adic fields, Dissertation, Florida State University (1965).Google Scholar