Residue fields of valued function fields of conics

Khanduja, Sudesh K. ; Usha , Garg (1993) Residue fields of valued function fields of conics Proceedings of the Edinburgh Mathematical Society, 36 (3). pp. 469-478. ISSN 0013-0915

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Related URL: http://dx.doi.org/10.1017/S0013091500018551

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.

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