On a query of Adrian Wadsworth (Q1125486)

The authors call a valued extension \((K,v)\) of the valued field \((K_0,v_0)\) residually transcendental, if the residue field \(k\) of \(v\) contains transcendental elements over the residue field \(k_0\) of \(v_0\). Let \(K=K_0(x,y)\) be a function field of a conic over \(K_0\) satisfying the equation \(x^2-cy^2=d\) for some \(c,d\) from \(K_0\). A necessary and sufficient condition is given for the existence of an extension \(v\) of \(v_0\) which is residually transcendental but not simple transcendental. As a corollary it is shown that in the above case there always exists a residually transcendental extension of \(v_0\) which is not simple transcendental. This answers a question of Adrian Wadsworth.