A Uniqueness Problem in Valued function Fields of Conics

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