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A uniqueness problem in simple transcendental extensions of valued fields

Published online by Cambridge University Press:  20 January 2009

Sudesh K. Khanduja
Sudesh K. Khanduja
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India
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Abstract

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Let υ0 be a valuation of a field K0 with value group G0 and υ be an extension of υ0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t∈K0(x)/K0 with υ(t) non-torsion mod G0 such that υ is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is “yes” if υ0 is henselian or if υ0 is of rank 1 with G0 a cofinal subset of the value group of υ in the latter case, and that it is “no” in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, υ)/(K0, υ0).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 1 , February 1994 , pp. 13 - 23
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Ax, J., A metamathematical approach to some problems in number theory, 1969 Number Theory Institute, Proc. Sympos. Pure Math. XX (1971), 161–190.Google Scholar
2.Bourbaki, N., Commutative Algebra, Chapter 6, Valuations (Hermann Publishers in Arts and Science, 1972).Google Scholar
3.Endler, O., Valuation Theory (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
4.Khanduja, S. K. and Garg, U., Rank 2 valuations of K(x), Mathematika 37 (1990), 97–105.CrossRefGoogle Scholar
5.Khanduja, S. K. and Garg, U., Stability theorem for valued function fields, submitted for publication.Google Scholar
6.Kuhlmann, F. V., Henselian function fields and tame fields, Manuscript 1990.Google Scholar
7.Matignon, M. and Ohm, J., A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104 (1988), 392–402.Google Scholar
8.Matignon, M. and Ohm, J., Simple transcendental extensions of valued fields III: The uniquenesss property, J. Math. Kyoto Univ. 30 (1990), 347–366.Google Scholar
9.Polzin, M., Prolongement de la valeur absolue de Gauss et probleme de Skolem, Bull. Soc. Math. France 116 (1988), 103–132.Google Scholar
10.Van der Waerden, B. L., Modern Algebra I (Ungar, New York, 1964).Google Scholar
11.Zariski, O. and Samuel, P., Commutative Algebra, Vol II (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar