Milnor K-theory
Matsumoto's presentation of K_2 of a field by symbols {a, b} is so clean that one is tempted to keep going: why not make symbols {a_1, ..., a_n} of any length and see what graded ring they generate? Milnor did exactly this, defining a graded ring built purely from symbols and two simple relations. The result is not the same as Quillen's K-theory in high degrees, but it is far more computable and turns out to be the part of K-theory most directly tied to quadratic forms and Galois cohomology.
For a field F, the Milnor K-theory K_*^M(F) is the quotient of the tensor algebra of the abelian group F^* (written additively as a Z-module, but with multiplicative origin) by the two-sided ideal generated by the Steinberg relations a ⊗ (1 - a) for a not equal to 0, 1. Explicitly, K_n^M(F) is generated by symbols {a_1, ..., a_n} of units, multilinear in each slot, with {a_1, ..., a_n} = 0 whenever a_i + a_j = 1 for some adjacent pair (equivalently a_i = 1 - a_j). By construction K_0^M(F) = Z, K_1^M(F) = F^*, and K_2^M(F) = K_2(F) by Matsumoto.
There is a natural ring map K_*^M(F) -> K_*(F) to Quillen K-theory that is an isomorphism in degrees 0, 1, 2 but not generally above. Milnor K-theory is the engine of two celebrated theorems: the Milnor conjecture (proved by Voevodsky) identifies K_n^M(F)/2 with the graded pieces of the Witt ring of quadratic forms and with Galois cohomology H^n(F, Z/2Z); the Bloch-Kato conjecture (Rost-Voevodsky) extends this to all primes, identifying K_n^M(F)/m with H^n(F, mu_m^{⊗n}).
For a finite field F_q, K_n^M(F_q) = 0 for n at least 2, because the bimultiplicative symbols all collapse. For the rationals, K_2^M(Q) = K_2(Q) decomposes via tame symbols into Z/2Z and the groups F_p^* over odd primes p, exactly matching the Quillen K_2.
Milnor K-theory of a finite field vanishes above degree 1.
Milnor introduced these groups in his 1970 paper on quadratic forms precisely to bridge K_2-symbols, quadratic forms, and Galois cohomology, and he conjectured the now-proved isomorphisms. The construction generalizes from fields to certain rings (Milnor-Witt K-theory and its variants), playing a role in motivic homotopy theory.