Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-06T02:38:40.124Z Has data issue: false hasContentIssue false
  • English
  • Français

On Extensions of Valuations with given Residue Field and Value Group

Part of: Field extensions

Published online by Cambridge University Press:  21 December 2009

Figen Öke
Figen Öke
Affiliation:
Trakya University, Department of Mathematics, 22030 Edirne, Turkey, E-mail: figenoke@gmail.com
Get access

Abstract

Let υ be a valuation on K with value group Gυ, residue field kυ, rank υ = t and K (x1, …, xn) be the field of rational functions over K with n variables. If G is the direct sum of G1 and d infinite cyclic groups where G1 is a totally ordered group containing Gυ as an ordered subgroup with [G1 : Gυ] < ∞ and k′ is a finite field extension of kυ then there exists a residual transcendental extension u of υ to K (x1, …, xn) such that rank u = t + d, Gu = G the algebraic closure of kυ in kυ is k′ and trans deg ku/kυ = n − d.

MSC classification

Secondary: 12F20: Transcendental extensions
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 191 - 197
Copyright
Copyright © University College London 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alexandru, V., Popescu, N. and Zaharescu, A., Minimal pairs of definition of a residual transcendental extension of a valuation. J. Math. Kyoto Univ. 30–32 (1990), 207–225.Google Scholar
2.Alexandru, V., Popescu, N. and Zaharescu, A., All valuations on K(x). J. Math. Kyoto Univ. 30–32 1991, 381–385.Google Scholar
3.Cohn, P. M., Algebraic Numbers and Algebraic Functions, Chapman and Hall (London, 1991).CrossRefGoogle Scholar
4.Khanduja, S. K., Value groups and simple transcendental extensions. Mathematika 38 1991, 381–385.CrossRefGoogle Scholar
5.Khanduja, S. K., Prolongations of valuations to simple transcendental extensions with given residue field and value group. Mathematika 38 1991, 386–390.Google Scholar
6.Matignon, M. and Ohm, J., A structure theorem for simple transcendental extensions of valued fields. Proc. Amer. Math. Soc. 104(2) 1988.CrossRefGoogle Scholar
7.Popescu, N. and Zaharescu, A., On a class of valuations on K(x). Anştiinţ. Univ. Ovidius Constanţa 2 1994, 120–136.Google Scholar