Steinberg symbol
K_2 of a field can feel abstract until you meet its building blocks. Given two nonzero elements a and b of a field, there is a recipe — using commuting diagonal matrices built from a and b inside the Steinberg group — that produces an element of K_2 called the Steinberg symbol {a, b}. These symbols behave like a strange multiplication table: they are bilinear in a and b, and they satisfy one curious vanishing rule. Matsumoto proved they generate all of K_2 of a field, and only these rules govern them.
Formally, for a field F and nonzero a, b in F, the Steinberg symbol {a, b} in K_2(F) is the commutator of certain lifts to St(F) of the diagonal matrices h_{ij}(a) and h_{ij}(b), where h_{ij}(u) = w_{ij}(u) w_{ij}(-1) and w_{ij}(u) = x_{ij}(u) x_{ji}(-u^{-1}) x_{ij}(u). The symbol is bimultiplicative: {a a', b} = {a, b}{a', b} and {a, b b'} = {a, b}{a, b'}. Matsumoto's theorem says K_2(F) is the abelian group on these symbols modulo the relations of bimultiplicativity and the Steinberg relation {a, 1 - a} = 1 for a not equal to 0, 1.
From the defining relations one deduces useful identities: {a, -a} = 1, hence {a, a} = {a, -1}, and {a, b} = {b, a}^{-1} (skew-symmetry). These symbols are the same gadgets that appear as Hilbert symbols and norm-residue symbols in number theory, which is why K_2 of a number field encodes reciprocity laws. The construction generalizes to Milnor K-theory, where symbols {a_1, ..., a_n} of higher length live in K_n^M(F).
In K_2(R) for the real numbers, {-1, -1} is the nonzero element of order 2, corresponding to the fact that K_2(R) has a Z/2Z summand. Over the rationals, the tame symbol at a prime p sends {a, b} to (-1)^{v(a)v(b)} a^{v(b)} b^{-v(a)} mod p, where v is the p-adic valuation.
{-1, -1} is the order-2 element of K_2(R).
The single relation {a, 1 - a} = 1 — called the Steinberg relation — is the deep input; everything else is formal. It is the algebraic shadow of the fact that the line through a and 1 - a in a certain geometry degenerates, and it is what links K_2 to the Bloch-Wigner dilogarithm and to motivic cohomology.