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The tag "spectral theory" is not obvious, but (very) relevant. Especially for (A).
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THC
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removed irrelevant tag, completed definitions (according to comments)
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YCor
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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.

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 $k$.

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.

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.

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THC
THC
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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 $k$.

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.