Residue fields of valued function fields of conics

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 36; no. 3; pp. 469 - 478
• Main Authors: Khanduja, Sudesh K., Garg, Usha
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.1993
• ISSN: 0013-0915; 1464-3839; 1464-3839
• DOI: 10.1017/S0013091500018551

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.