Grothendieck Rings of $mathbb{Z}$-Valued Fields

Bulletin of Symbolic Logic 7 (2):262-269 (2001)

Abstract
We prove the triviality of the Grothendieck ring of a $\mathbb{Z}$-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 specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point
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