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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

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