Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T04:05:38.909Z Has data issue: false hasContentIssue false

Finite Extensions of Valued Fields

Published online by Cambridge University Press:  20 November 2018

Seth Warner*
Affiliation:
Duke University, Durham, NC 27706
Rights & Permissions [Opens in a new window]

Abstract

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.

A corollary of the main result is that if L is a finite-dimensional Galois extension of a field K and if w is a valuation of L extending a valuation v of K, then K is closed in L if and only if all valuations of L extending v are dependent. A further consequence is a generalization of Ostrowski's criterion for a real-valued valuation to be henselian.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bourbaki, N., Algèbre Commutative, Ch. 5—6, Hermann, Paris, 1964.Google Scholar
2. Krasner, M., Théorie non abélienne des corps de classes pour les extensions finies et separable s des corps values complets; principes fondamentaux; espaces de polynômes et transformation T; lois d'unicité, d'ordination et d'existence, C.R. Acad. Sci. Paris 222 (1946), pp. 626628.Google Scholar
3. Nachbin, L., On strictly minimal topological division rings, Bull. Amer. Math. Soc. 55 (1949), pp. 11281136.Google Scholar
4. Nagata, M., On the theory of Henselian rings, Nagoya Math. J. 5 (1953), pp. 45—57.Google Scholar
5. Ostrowski, A., Uber sogenannte perfekte Kôrper, J. Reine Angew. Math. 147 (1917), pp. 191—204.Google Scholar
6. Ostrowski, A., Untersuchungen zur arithmetischen Théorie der Körper, Math. Z. 39 (1935), pp. 269404.Google Scholar
7. Rigo, T. and Warner, S., Topologies extending valuations, Canad. J. Math. 30 (1978), pp. 164—169.Google Scholar
8. Weber, H., Topologische Charakterisierung globaler Kôrper und algebraischer Funktionenkôrper in einer Variablen, Math. Z. 169 (1979), pp. 167177.Google Scholar