K_2

K_1 forgot the elementary matrices; K_2 asks a deeper question: when you build the identity matrix out of elementary moves, in how many genuinely different ways can you do it? Elementary matrices satisfy obvious relations (the Steinberg relations), and any sequence of moves returning you to the identity that follows only from those obvious relations is uninteresting. K_2 is the group of the remaining, non-obvious relations — the secret syzygies among elementary matrices. It is the first K-group whose definition is invisible at the level of modules and matrices alone.

Concretely, one forms the Steinberg group St(R) by taking generators x_{ij}(r) for i not equal to j, mimicking the elementary matrices e_{ij}(r), subject precisely to the Steinberg relations they are known to satisfy. There is a surjection St(R) -> E(R). By Milnor's definition, K_2(R) is the kernel of this map: K_2(R) = ker(St(R) -> E(R)). It turns out St(R) is the universal central extension of E(R), and K_2(R) is its center, hence an abelian group and the Schur multiplier of E(R).

For a field F, Matsumoto's theorem computes K_2(F) completely: it is generated by Steinberg symbols {a, b} for units a, b, subject to bimultiplicativity and the single Steinberg relation {a, 1 - a} = 1 whenever a and 1 - a are both nonzero. This presentation ties K_2 directly to reciprocity laws in number theory and to the symbols appearing in class field theory; for instance K_2(Q) decomposes via tame symbols at each prime.