K-theory of a field

Fields are the simplest rings, so their K-groups are the natural first laboratory: every finitely generated module is a vector space, every projective is free, and the low K-groups reduce to familiar arithmetic invariants. Yet even here the higher K-groups of a field reach into the deepest waters of number theory, encoding units, Brauer classes, étale cohomology, and special values of zeta functions. The K-theory of a field is small at the bottom and vast above.

For a field F: K_0(F) = Z (every vector space is determined by dimension); K_1(F) = F^*, the multiplicative group of units; and K_2(F) is generated by Steinberg symbols {a, b} by Matsumoto's theorem. There is a natural surjection from Milnor K-theory K_n^M(F) to Quillen K_n(F); it is an isomorphism for n at most 2 but not in general. The torsion in K_*(F) is governed by the Bloch-Kato conjecture (now the Rost-Voevodsky theorem), which identifies the mod-m Milnor K-theory with Galois cohomology.

Two appearances stand out. The 2-torsion and symbol structure of K_2 and Milnor K-theory control the Brauer group: K_2(F)/2 surjects onto the 2-torsion of Br(F) via symbols of quaternion algebras. And for global and local fields the higher K-groups carry arithmetic: K_*(F_q) is finite cyclic in odd degrees, K_*(Q) and K_*(number field) relate to zeta values by the Quillen-Lichtenbaum conjecture, also resolved through the Bloch-Kato machinery. Thus the K-theory of a field is a meeting point of algebra, arithmetic, and motivic cohomology.