Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes and consider their classes $[\mathrm{Spec}(K)]$ and $[\mathrm{Spec}(K')]$ in $K_0(V_k)$.

I have two general questions:

(A) what are the most interesting properties/criteria which lead to the equality $[\mathrm{Spec}(K)] = [\mathrm{Spec}(K')]$ ?

(B) when can we decide that $[\mathrm{Spec}(K)] \ne [\mathrm{Spec}(K')]$ ?

I understand that some of the first properties will be pretty easy or expected (but still interesting to mention), but also that the "next generation" of properties could be highly interesting.

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices. Putting this tag just spams people not interested with this stuff... $\endgroup$ – YCor Mar 12 at 0:34