Field extensions in Grothendieck rings 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.   
 A: 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
