For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$.
What information of $F$ can be recovered from the graded ring $K^M(F)$? Can it determine $F$ uniquely? For instance, we can consider $F$ among local fields or number fields.
Is there any example of two non-isomorphic fields $F_i$ such that $K^M(F_1) \cong K^M(F_2)$? There is no such example if $F_i$ are both finite or algebraically closed.
Motivations:
(1) Number fields can be determined by their absolute Galois groups.
(2)This paper proves that the function field of an algebraic variety of dimension ≥ 2 over an algebraically closed field is completely determined by its first and second Milnor K-groups;
(3) There are different $p$-adic fields with isomorphic $K_1$.
(4) Norm residue isomorphism theorem.
