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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
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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

Information

Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 191 - 197
Copyright
Copyright © University College London 2009

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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