On residually transcendental valued function fields of conics

Abstract

Let K/Ko be a finitely generated field extension of transcendence degree 1. Let u⁰ be a valuation of K₀ and v a valuation of K extending v₀ such that the residue field of vis a transcendental extension of the residue field k⁰ of v⁰/such a prolongation vwill be called a residually transcendental prolongation of v⁰. Byan element with the uniqueness propertyfor (K, v)/(K⁰, v⁰) (or more briefly for v/v⁰)we mean an element / of K having u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of v onto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over k₀; that is v coincides with the Gaussian valuation v^t_o on the subfield K⁰(t) defined by v₀ⁱ(∑_iα_it ⁱ)=min_i(v₀(α_i)); (ii) vis the only valuation of K (up to equivalence) extending the valuation v^t_o.