The Galois Symbol and Norm Residue Map

The norm residue homomorphism (or Galois symbol) connects Milnor K-theory to Galois cohomology. This map is the central object of the Bloch--Kato conjecture, now a theorem of Voevodsky and Rost, which provides a complete understanding of the relationship between symbols in K-theory and etale cohomology classes.

The Kummer map and Galois cohomology

Definition4.4Kummer theory and the norm residue symbol

For a field $F$ containing a primitive $n$ -th root of unity $\zeta_n$ (with $\operatorname{char}(F) \nmid n$ ), the Kummer map is

$\partial: F^{\times} \to H^1(G_F, \mu_n), \quad a \mapsto (\sigma \mapsto \sigma(\sqrt[n]{a}) / \sqrt[n]{a})$

where $G_F = \operatorname{Gal}(F^{\mathrm{sep}}/F)$ is the absolute Galois group. This gives an isomorphism $F^{\times} / (F^{\times})^n \xrightarrow{\sim} H^1(G_F, \mu_n)$ (Kummer theory).

The norm residue homomorphism (or Galois symbol) is the degree- $m$ extension:

$h_n^m: K_m^M(F) / n \to H^m(G_F, \mu_n^{\otimes m})$

defined by

$h_n^m(\{a_1, \ldots, a_m\}) = \partial(a_1) \cup \cdots \cup \partial(a_m)$

where $\cup$ is the cup product in Galois cohomology.

RemarkWell-definedness

To verify $h_n^m$ is well-defined, one must check the Steinberg relation: for $a \in F \setminus \{0, 1\}$ ,

$\partial(a) \cup \partial(1-a) = 0 \in H^2(G_F, \mu_n^{\otimes 2}).$

The class $\partial(a) \cup \partial(1-a)$ corresponds to a central simple algebra (the cyclic algebra $(a, 1-a)_n$ ), and the Steinberg relation asserts this algebra is split. This can be verified: $F(\sqrt[n]{a})$ contains an $n$ -th root of $1-a$ (since $1 - a = 1 - a$ and the extension $F(\sqrt[n]{a})/F$ splits the algebra).

Low-degree cases

ExampleDegree 1: Kummer theory

For $m = 1$ : $h_n^1: K_1^M(F)/n = F^{\times}/(F^{\times})^n \to H^1(G_F, \mu_n)$ is the Kummer isomorphism. This is classical and holds for all $n$ with $\operatorname{char}(F) \nmid n$ .

For $F = \mathbb{Q}$ and $n = 2$ : $\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^2 \cong H^1(G_\mathbb{Q}, \mu_2)$ . The group $\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^2$ has generators $\{-1, 2, 3, 5, 7, \ldots\}$ (sign and primes), and each corresponds to a quadratic extension $\mathbb{Q}(\sqrt{d})$ .

ExampleDegree 2: Brauer group

For $m = 2$ : $h_n^2: K_2^M(F)/n \to H^2(G_F, \mu_n^{\otimes 2})$ . When $F$ contains $\mu_n$ , we have $\mu_n^{\otimes 2} \cong \mu_n$ and $H^2(G_F, \mu_n) \cong \operatorname{Br}(F)[n]$ (the $n$ -torsion in the Brauer group).

The map sends a symbol $\{a, b\}$ to the class of the cyclic algebra $(a, b)_n$ : the $F$ -algebra generated by $u, v$ with $u^n = a$ , $v^n = b$ , $vu = \zeta_n uv$ .

The Merkurjev--Suslin theorem (1982): $h_n^2$ is an isomorphism for all fields and all $n$ . This was a breakthrough, implying:

$\operatorname{Br}(F)[n]$ is generated by cyclic algebras (when $\mu_n \subset F$ ).
$K_2(F)/n \cong H^2_{\text{et}}(F, \mu_n^{\otimes 2})$ .

The Bloch--Kato conjecture

Theorem4.1Bloch--Kato Conjecture (Voevodsky--Rost)

For any field $F$ and any positive integer $n$ with $\operatorname{char}(F) \nmid n$ , the norm residue homomorphism

$h_n^m: K_m^M(F)/n \xrightarrow{\;\sim\;} H^m(G_F, \mu_n^{\otimes m})$

is an isomorphism for all $m \geq 0$ .

RemarkHistory and proof

The proof of the Bloch--Kato conjecture was achieved through the work of many mathematicians over several decades:

$m = 1$ : Classical (Kummer theory).
$m = 2$ : Merkurjev--Suslin (1982) for all $n$ ; Merkurjev (1981) for $n = 2$ .
$n = 2$ , all $m$ : Voevodsky (1996--2003), using motivic cohomology and the Steenrod algebra. This is the Milnor conjecture, for which Voevodsky received the Fields Medal.
$n$ odd prime, all $m$ : Voevodsky with Rost's chain lemma (2008--2011). The proof uses:
- Motivic cohomology and its relationship to Milnor K-theory ( $K_m^M(F) \cong H^m_{\text{mot}}(F, \mathbb{Z}(m))$ ).
- The motivic Steenrod algebra and its operations.
- Norm varieties and Rost's degree formula.
- Reduction steps via Rost's chain lemma.
General $n$ : Follows from the prime power cases by a formal argument.

Consequences

ExampleConsequences of Bloch--Kato

The Bloch--Kato conjecture (theorem) has profound consequences:

Galois cohomology computation: $H^m(G_F, \mu_n^{\otimes m})$ is computed by explicit symbols $\{a_1, \ldots, a_m\}$ . No "exotic" cohomology classes exist.
Quadratic forms: Milnor's conjecture relating $K_*^M(F)/2$ , the graded Witt ring $I^*/I^{*+1}$ , and $H^*(F, \mathbb{Z}/2)$ follows as a corollary.
Higher class field theory: For a local field $F$ of residue characteristic $p$ and $\ell \neq p$ :

$K_n^M(F)/\ell \cong H^n(G_F, \mu_\ell^{\otimes n}) \cong \begin{cases} \mathbb{Z}/\ell & n = 0, 1 \\ \mu_\ell(F) \cong \mathbb{Z}/\gcd(\ell, q-1) & n = 2 \\ 0 & n \geq 3 \end{cases}$

where the vanishing for $n \geq 3$ follows from the cohomological dimension $\operatorname{cd}_\ell(G_F) = 2$ .

Lichtenbaum conjectures: The etale K-theory of number rings is related to special values of zeta functions, with the Bloch--Kato conjecture providing the comparison between algebraic and etale K-theory.