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. 



Field extensions in Grothendieck rings
In characteristic zero $[\mathrm{Spec}(K)] = [\mathrm{Spec}(K')]$ for finite field extensions of $k$ implies that $K$ and $K'$ are isomorphic.


Indeed, by the Larsen-Lunts theorem for smooth projective connected schemes of finite type $[X] = [Y]$ implies that $X$ and $Y$ are stably birational; this applies to $\mathrm{Spec}(K)$, $\mathrm{Spec}(K')$ as they are connected smooth and projective. Now if $\mathrm{Spec}(K)$, $\mathrm{Spec}(K')$ are stably birational, then they are isomorphic. This is because if $X$ is smooth projective and stably birational to $\mathrm{Spec}(K)$, then $K = \Gamma(X, \mathcal{O}\_X)$. Same argument applies to products of fields that is to reduced zero-dimensional finite $k$-schemes: here the Larsen-Lunts theorem will match up the stable birational types of the connected components.


In characteristic $p > 0$ the result may be still true but hopeless to prove without resolution of singularities.


UPDATE: the result above about fields is given in Proposition 5 in a paper by Liu and Sebag where they study what $[X] = [Y]$ implies in general in characteristic zero, using Larsen-Lunts Theorem: <https://link.springer.com/article/10.1007/s00209-009-0518-7>


