Grothendieck Rings of ℤ-Valued Fields

Abstract We prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K . We construct a definable bijection from the plane K 2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.