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 ?


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