A remark about the norm-residue homomorphism \(h: K_*F/2\rightarrow H^*(F,\mathbb{Z}/2)\) (Q1423788)

Let \(F\) be a field of characteristic \(\neq 2\) with separable closure \(F_s\), let \(K_nF/2\) be the \(n\)-th Milnor \(K\)-group modulo \(2\), and let \(H^n(F)=H^n(\text{Gal}(F_s/F),{\mathbb Z}/2)\) be the \(n\)-th Galois cohomology group with coefficients in \({\mathbb Z}/2\). The purpose of the present nicely written paper is to give an elementary and self-contained proof of the fact that there is a well-defined group homomorphism \(h_n : K_nF/2 \to H^n(F)\) mapping a symbol \(\{a_1,\ldots ,a_n\}\in K_nF/2\) to the \(n\)-fold cup product \((a_1)\cup\cdots \cup (a_n)\in H^n(F)\). The major part of the proof concerns the fact that the cup product \((a)\cup (b)\) vanishes if and only if \(b\) is a norm of the extension \(F(\sqrt{a})/F\). One part of the Milnor conjecture (now a theorem due to Voevodsky) states that this map \(h_n\) is indeed an isomorphism for all \(n\). The paper concludes with some comments on quadratic forms and in particular the part of the Milnor conjecture concerning the graded Witt ring, and on the literature (without mention of Voevodsky's work).