## Bulletin of Symbolic Logic

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

#### 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.

#### Article information

Source
Bull. Symbolic Logic, Volume 7, Number 2 (2001), 262-269.

Dates
First available in Project Euclid: 20 June 2007

https://projecteuclid.org/euclid.bsl/1182353778

Mathematical Reviews number (MathSciNet)
MR1839548

Zentralblatt MATH identifier
0988.03058

JSTOR
Cluckers, Raf; Haskell, Deirdre. Grothendieck Rings of $\mathbb{Z}$-Valued Fields. Bull. Symbolic Logic 7 (2001), no. 2, 262--269. https://projecteuclid.org/euclid.bsl/1182353778