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. Suppose that either ( K 0 , v 0 ) is henselian or v 0 is of rank 1, the algebraic 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 there exists an explicitly constructible element t in K , with v ( t ) non‐torsion modulo G 0 such that the valuation of K 0 ( t ), obtained by restricting v , has a unique extension to K . This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K 0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [ 8 ].