Matsumoto's theorem

Remarks

In the language of Milnor K-theory, the theorem says that the natural map $K_2^M(F) \to K_2(F)$ is an isomorphism. The symbol $\set{a, b}$ is defined in the Steinberg group as a commutator of lifts of the diagonal matrices $\operatorname{diag}(a, a^{-1}, 1)$ and $\operatorname{diag}(b, 1, b^{-1})$. The hard part of the proof is the absence of other relations. From a Steinberg symbol one builds a central extension of $SL(F)$, and the universal property of the Steinberg group gives the homomorphism $h$. Matsumoto proved a more general result for the universal central extensions of split simply connected groups over fields.

Example: by the theorem, the real Hilbert symbol induces $h \colon K_2(\R) \to \set{\pm 1}$ with $h(\set{-1, -1}) = -1$, so $\set{-1, -1} \ne 1$ in $K_2(\R)$.

Going deeper

For a student

The theorem turns classical symbols into homomorphisms out of $K_2$: the Hilbert symbol, the norm residue symbol and the tame symbol all factor through $K_2(F)$. This is how one computes $K_2(\Q)$ with tame symbols and relates $K_2(F)$ to the Brauer group. It is also the starting point of Milnor K-theory, which takes the same generators and relations in every degree.