A UNIQUENESS PROBLEM IN VALUED FUNCTION-FIELDS OF CONICS

Let v(0) be a valuation of a field K-0 with value group G(0). Let K be a function field of a conic over K-0, and let v be an extension of v( 0) to K with value group G such that G/G(0) is not a torsion group. Su ppose that either (K-0, v(0)) is henselian or v(0) is of rank 1, the a lgebraic closure of K-0 in K is a purely inseparable extension of K-0, and G(0) is a cofinal subset of G. In this paper, it is proved that t here exists an explicitly constructible element 1 in K, with v(t) non- torsion module G(0) such that the valuation of K-0(t), obtained by res tricting v, has a unique extension to K. This generalizes the result p roved by Khanduja in the particular case, when K is a simple transcend ental extension of K-0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].