On residually transcendental valued function fields of conics

Khanduja, Sudesh K. (1996) On residually transcendental valued function fields of conics Glasgow Mathematical Journal, 38 (2). pp. 137-145. ISSN 0017-0895

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Related URL: http://dx.doi.org/10.1017/S0017089500031360

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 vto on the subfield K0(t) defined by v0i(∑iαit i)=mini(v0i)); (ii) vis the only valuation of K (up to equivalence) extending the valuation vto.

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