Milnor conjecture

Remarks

Milnor stated this in 1970, and Voevodsky proved it using motivic cohomology and motivic Steenrod operations. Before that, the case $n \le 1$ followed from Kummer theory, the case $n = 2$ was proved by Merkurjev, and the case $n = 3$ by Merkurjev–Suslin and, independently, by Rost. It is the case $\ell = 2$ of the norm residue isomorphism theorem.

Example: $F = \R$. Since $\R^\times / (\R^\times)^2 = \set{\pm 1}$, the group $K_n^M(\R)/2$ is generated by $\set{-1, \ldots, -1}$. The cohomology ring $H^*(\R, \Z/2\Z)$ is $\F_2[x]$ with $x = \delta(-1)$, so the symbol maps to $x^n \ne 0$, and both sides are $\Z/2\Z$.

The hypothesis on the characteristic is needed for the Kummer map. In characteristic $2$, squaring is not a surjective map of $(F^s)^\times$ with kernel $\mu_2$, and the Galois symbol is not defined in this form.

Going deeper

For a student

Combined with the work of Orlov, Vishik and Voevodsky, it also gives the Milnor conjecture on quadratic forms. That result says that the graded ring of the Witt ring of $F$ for its fundamental ideal is isomorphic to $K_*^M(F)/2$. So quadratic forms over $F$ are classified, up to a filtration, by Galois cohomology invariants.