# An inverse problem for Grothendieck rings of varieties

Suppose $$A$$ is a given commutative ring, and suppose that one knows that $$A$$ is isomorphic to the Grothendieck ring of $$k$$-varieties for some unknown field $$k$$.

Can $$k$$ be recovered from $$A$$ ? If not, what about the characteristic of $$k$$ ?

A related question is obviously the following: if $$k$$ and $$k'$$ are nonisomorphic fields, can the Grothendiek rings $$K_0(V_k)$$ and $$K_0(V_{k'})$$ be isomorphic ?