Valued fields with a total residue map

When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, inModel Theory of Algebra and Arithmetic,Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin,1979)] observed that the total residuemap res :k((t))-> k, which picks out the constantterm of the Laurent series, is definable in the language of rings with a parameter fort.Driven by this observation, we study the theory VFres,iota of valued fields equipped with a linear form res :K -> k which restricts to the residue map on the valuation ring.We prove that VFres,iota does not admit a model companion. In addition, we show that (k((t)),res) is undecidable when ever k is an infinite field. As a consequence, we get that(C((t)),Res0) is undecidable, where Res0: f -> Res0(f)maps f to its complex residue at 0.