Grothendieck rings of Laurent series fields

We study Grothendieck rings (in the sense of model theory) of fields, extending previous work of Haskell and the author in [R. Cluckers, D. Haskell, Bull. Symbolic Logic 7 (2) (2001) 262–269]. We construct definable bijections from the line to the line minus one point in the language of rings for valued fields like fields of formal Laurent series over p-adic numbers and fields of formal Laurent series over local fields of strictly positive characteristic. It follows that the Grothendieck rings of these fields are trivial.