Grothendieck rings of Laurent series fields

Abstract

We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields, for example, fields of Laurent series over the real numbers, over p-adic numbers and over finite fields, and construct definable bijections from the line to the line minus one point.