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A Theorem on Valuation Rings and its Applications

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Harvard University and Kyoto University
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In the present paper, we first prove the following

THEOREM 1. Let K be a field, x a transcendental element over K, and V* a valuation ring of K(x). Set V= V* ∩ K. Denote by p* and p the maximal ideals of V* and V respectively. If (i) V*/p* is not algebraic over V/p and (ii) the value group of V* is isomorphic to Zn (Z=the module of rational integers), i.e., V* is of rank n and discrete in the generalized sense, then V*/p* is a simple transcendental extension of a finite algebraic extension of V/p.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 29 , March 1967 , pp. 85 - 91
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

References

[1] Abhyankar, S., On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348.CrossRefGoogle Scholar
[2] Igusa, J., On a theorem of Lüroth, Mem. Univ. Kyoto 26 (1950-51), 251–253.Google Scholar
[3] Samuel, P., Some remarks on Ltiroth’s theorem, Mem. Univ. Kyoto 27 (1952-53), 223–224.Google Scholar
[4] Segre, B., Sur un problème de M. Zariski, Colloque internat, d’algèbre et de théorie de nombres, Paris 1949, 135–138.Google Scholar
[5] Weber, H., Lehrbuch der Algebra, II, Braunschweig, 1899.Google Scholar