Merkurjev–Suslin theorem

Remarks

Without the assumption $\zeta \in F$, the theorem says that the Galois symbol $K_2(F)/n \to H^2(F, \mu_n^{\otimes 2})$ is an isomorphism for every $n$ invertible in $F$. When $\zeta \in F$, the module $\mu_n^{\otimes 2}$ is isomorphic to $\mu_n$, and $H^2(F, \mu_n) \cong \operatorname{Br}(F)[n]$ by Kummer theory and Hilbert's Theorem 90, which gives the form above. The proof, by Merkurjev and Suslin, uses the K-theory of Severi–Brauer varieties and Hilbert's Theorem 90 for $K_2$.

Consequence: suppose $F$ contains a primitive $n$-th root of unity. Then every central simple algebra over $F$ whose class has order dividing $n$ is Brauer equivalent to a tensor product of symbol algebras. For $n = 2$ this says that such an algebra is Brauer equivalent to a tensor product of quaternion algebras, a result proved earlier by Merkurjev.

Example: $F = \R$, $n = 2$. Here $K_2(\R)/2 \cong \Z/2\Z$ is generated by $\set{-1, -1}$, and $\operatorname{Br}(\R)[2] = \operatorname{Br}(\R) \cong \Z/2\Z$ is generated by the quaternions, the image of $\set{-1, -1}$.

Going deeper

For a student

This is the degree two case of the norm residue isomorphism theorem, which identifies Milnor K-theory modulo $n$ with Galois cohomology in all degrees. It answered an old question on the generation of the Brauer group by cyclic algebras, under the root of unity hypothesis. It also gives the first nontrivial case of the Milnor conjecture, for $n = 2$ in degree two.