Algebraic K-theory - Milnor K-theory

Milnor K-theory

The above expression for K₂ of a field k led Milnor to the following definition of "higher" K-groups by

,

thus as graded parts of a quotient of the tensor algebra of the multiplicative group k× by the two-sided ideal, generated by the

For n = 0,1,2 these coincide with those below, but for n≧3 they differ in general. For example, we have KM
n(F_q) = 0 for n≧3. Milnor K-theory modulo 2 is related to étale (or Galois) cohomology of the field by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.