Milnor conjecture on quadratic forms

Remarks

Milnor defined the maps $s_n$ in 1970 and asked whether they are isomorphisms. Existence and surjectivity are elementary: the Steinberg relation holds because $\langle\langle a, 1 - a \rangle\rangle$ is isotropic, hence hyperbolic, and $I^n(F)$ is generated by $n$-fold Pfister forms. Injectivity is the hard part, proved by Orlov, Vishik and Voevodsky using motivic cohomology. For $n = 1$ the statement is the classical isomorphism $F^\times/(F^\times)^2 \cong I(F)/I^2(F)$ given by the discriminant. For $n = 2$ it follows from Merkurjev's theorem on the $2$-torsion of the Brauer group, through the Clifford invariant. Example: for $F = \R$, both sides are $\Z/2$ for every $n$, generated by $\set{-1, \ldots, -1}$ and by the class of $\langle 1, 1, \ldots, 1 \rangle$ of dimension $2^n$. Combined with Voevodsky's theorem $K^M_n(F)/2 \cong H^n(F, \Z/2)$, it identifies $I^n(F)/I^{n+1}(F)$ with Galois cohomology.

Going deeper

For a student

The filtration of the Witt ring by powers of the fundamental ideal organizes the classical invariants of quadratic forms. The dimension modulo $2$, the discriminant and the Clifford invariant are the first three steps, and the theorem provides an invariant in every degree. It is closely tied to the Milnor conjecture on Galois cohomology, proved by Voevodsky, and both are parts of what Milnor proposed in his 1970 paper. The Arason–Pfister Hauptsatz implies that the intersection of all $I^n(F)$ is zero, so a form is determined by its images in these layers.