On a query of Adrian Wadsworth

Abstract

Lei v₀ be a valuation of a field K₀ with residue field k₀ of characteristic #2. Suppose that K is a function field of a conic over K₀ and v is a prolongation of v₀ to K. whose residue field k is not algebraic over k₀. It is well known (cf [Residue fields of valued function fields of conies, Proc. Edinb. math. Sac 36 (1993), 469-78]) that either k is a simple transcendental extension of a finite extension of k₀,or k is a regular function field of a conic over k₀. The paper contains the answer to a natural question posed by Wadsworth which states that if v₁, v₁ are any two prolongations of v₀ to K with residue fields k₁ k₂ non-algebraic over k₀ such that neither k₁, or k₂ is a simple transcendental extension of a finite extension of k₀, is it true that k₁, is t₀-isomorphic to k_T.