Norm residue isomorphism theorem

Remarks

The map is well defined because the cup product $\delta(a) \cup \delta(1 - a)$ vanishes for $a \ne 0, 1$. For $n = 0$ the statement is trivial, and for $n = 1$ it is Kummer theory with Hilbert's Theorem 90: $K^\times/(K^\times)^\ell \cong H^1(K, \mu_\ell)$. For $n = 2$ it is the Merkurjev–Suslin theorem, which identifies $K_2(K)/\ell$ with the $\ell$-torsion of the Brauer group when $K$ contains the $\ell$-th roots of unity. The case $\ell = 2$ is the Milnor conjecture, proved by Voevodsky; the general case, formerly the Bloch–Kato conjecture, was completed by Voevodsky with essential contributions of Rost. The proof uses motivic cohomology and is far beyond elementary methods. The theorem also holds with $\ell$ replaced by any positive integer $m$ invertible in $K$, with $\mu_m^{\otimes n}$ in place of $\mu_\ell^{\otimes n}$.

Going deeper

For a student

In degree $2$, suppose $K$ contains the $\ell$-th roots of unity. Then every central simple algebra of exponent $\ell$ is Brauer equivalent to a tensor product of cyclic algebras. For $\ell = 2$ it yields, with further work, the description of the graded Witt ring of quadratic forms by Galois cohomology, also called the Milnor conjecture on quadratic forms.