---
tag: 5Q5VN56QH4
kind: statement
role: theorem
name: Norm residue isomorphism theorem
msc: [19D45, 12G05]
uses: [2YXDQ7Q445, EESNTXFWHZ, JJ91A1V8NQ, 0SN0MAEKYX, 1ZB13N78XW]
about: [2YXDQ7Q445, 9MPCZ3EAJM]
sources:
  - url: https://en.wikipedia.org/wiki/Norm_residue_isomorphism_theorem
same_as:
  wikidata: Q7051622
  wikipedia: Norm residue isomorphism theorem
---

## Statement

Let $K$ be a field, let $\ell$ be a [[prime number|0SN0MAEKYX]] different from the [[characteristic|1ZB13N78XW]] of $K$, and let $n \ge 0$.
Let $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$, and let $\mu_\ell \subseteq K^s$ be the group of $\ell$-th roots of unity.
Let $\mu_\ell^{\otimes n}$ be its $n$-fold tensor power over $\Z$, with $\mu_\ell^{\otimes 0} = \Z/\ell\Z$.
Let $\delta \colon K^\times \to H^1(K, \mu_\ell)$ be the Kummer map.
Then the norm residue map
$$K_n^M(K)/\ell \longrightarrow H^n(K, \mu_\ell^{\otimes n}), \qquad \set{a_1, \ldots, a_n} \longmapsto \delta(a_1) \cup \cdots \cup \delta(a_n),$$
from [[Milnor K-theory|2YXDQ7Q445]] modulo $\ell$ to [[Galois cohomology|EESNTXFWHZ]], is well defined and is an isomorphism of groups.

## 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|YFF8B5A0QG]]: $K^\times/(K^\times)^\ell \cong H^1(K, \mu_\ell)$.
For $n = 2$ it is the Merkurjev–Suslin theorem, which identifies $K_2(K)/\ell$ with the $\ell$-torsion of the [[Brauer group|BEW48TA9WZ]] 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 $\mu_\ell^{\otimes n}$.

## Explanations

### L1 human

Milnor K-groups are defined by simple generators and relations, while Galois cohomology groups are defined through the symmetries of field extensions.
The theorem says that, modulo a prime $\ell$, these two very different constructions give the same groups.
So questions about Galois cohomology can be answered with explicit symbols, and the other way round.

### L2 human

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.