Residue fields of valued function fields of conics

Abstract

Suppose that K is a function field of a conic over a subfield K₀. Let v₀ be a valuation of K₀ with residue field k₀ of characteristic ≠2. Let v be an extension of v₀ to K having residue field k. It has been proved that either k is an algebraic extension of k₀ or k is a regular function field of a conic over a finite extension of k₀. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.