A uniqueness problem in valued function fields of conics

Abstract

Let v₀ be a valuation of a field K₀ with value group G₀. Let K be a function field of a conic over K₀, and let v be an extension of v₀ to K with value group G such that G/G₀ is not a torsion group. Suppose that either (K₀, v₀) is henselian or v₀ is of rank 1, the algebraic closure of K₀ in K is a purely inseparable extension of K₀, and G₀ is a cofinal subset of G. In this paper, it is proved that there exists an explicitly constructible element t in K, with v(t) non-torsion modulo G₀ such that the valuation of K₀(t), obtained by restricting v, has a unique extension to K. This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K₀ (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].