On quadratic forms and Galois cohomology (Q915765)

This is a survey article concerning the ``Milnor problems'' on the isomorphism of the groups \(\bar I^ nF=I^ nF/I^{n-1}F,\quad H^ nF,\quad H^ n_ qF,\quad k_ nF\) for all fields F and all \(n\in {\mathbb{N}}_ 0\). It was presented at the 1986 Corvallis Conference on Quadratic Forms and Real Algebraic Geometry. In the meantime considerable progress has been achieved by \textit{B. Jacob} and \textit{M. Rost} [Invent. Math. 96, 551-570 (1989; Zbl 0685.10015)] and, independently, by Merkur'ev-Suslin: For \(n\leq 3\) these groups are all isomorphic, for \(n=4\) the homomorphism \(e^ 4_ F: \bar I^ 4F\to H^ 4F\) exists.