Milnor conjecture

Let $F$ be a field with $\operatorname{char} F \neq 2$.

Define the Milnor $K$-ring of $F$ as the graded ring

$$K_*^M(F) \;=\; \Big(\bigoplus_{n\ge 0} (F^\times)^{\otimes n}\Big)\Big/ I,$$

where the tensor algebra is taken over $\mathbb{Z}$ on the abelian group $F^\times$ (so that the degree-$0$ part is $\mathbb{Z}$ and the degree-$1$ part is $F^\times$), and $I$ is the two-sided homogeneous ideal generated by the elements $a \otimes (1-a)$ with $a \in F^\times$, $a \neq 1$. Write $\{a_1,\dots,a_n\} \in K_n^M(F)$ for the class of $a_1 \otimes \cdots \otimes a_n$, and $K_n^M(F)/2 = K_n^M(F)/2K_n^M(F)$.

Let $H^n_{\text{ét}}(F,\mathbb{Z}/2\mathbb{Z}) = H^n(\operatorname{Gal}(F_{\mathrm{sep}}/F), \mathbb{Z}/2\mathbb{Z})$ denote the Galois (equivalently étale) cohomology of $F$ with coefficients in the trivial module $\mathbb{Z}/2\mathbb{Z}$, where $F_{\mathrm{sep}}$ is a separable closure of $F$. Since $\operatorname{char} F \neq 2$, Kummer theory provides an isomorphism

$$F^\times/(F^\times)^2 \;\xrightarrow{\ \sim\ }\; H^1_{\text{ét}}(F,\mathbb{Z}/2\mathbb{Z}), \qquad a \mapsto (a),$$

and the cup product on Galois cohomology makes the assignment

$$\{a_1,\dots,a_n\} \;\longmapsto\; (a_1)\cup (a_2)\cup \cdots \cup (a_n)$$

a well-defined homomorphism of graded rings, the norm residue map

$$h_n \colon K_n^M(F)/2 \;\longrightarrow\; H^n_{\text{ét}}(F,\mathbb{Z}/2\mathbb{Z}),$$

with $h_0$ the reduction $\mathbb{Z}/2 \to H^0_{\text{ét}}(F,\mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$.

Then $h_n$ is an isomorphism

$$K_n^M(F)/2 \;\cong\; H^n_{\text{ét}}(F,\mathbb{Z}/2\mathbb{Z})$$

for every integer $n \ge 0$ and every field $F$ of characteristic different from $2$.