The calculation of K2 of a field F led Milnor to the following ad hoc definition of "higher" K-groups by
For example, we have for n ≧ 2; is an uncountable uniquely divisible group; is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; is the direct sum of the multiplicative group of and an uncountable uniquely divisible group; is the direct sum of the cyclic group of order 2 and cyclic groups of order for all odd prime .
Milnor K-theory plays a fundamental role in higher class field theory, replacing in the one-dimensional class field theory.
Milnor K-theory modulo 2, denoted k*(F) is related to étale (or Galois) cohomology of the field F by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch–Kato conjecture, proved by Voevodsky, Rost, and others.
There are homomorphisms from kn(F) to the Witt ring of F by taking the symbol
where the image is a Pfister form of dimension 2n. The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In. The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.
- Lam (2005) p.366
- Gille & Szamuely (2006) p.184
- Lam (2005) p.316
- Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
- Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
- Milnor, John Willard (1970), With an appendix by J. Tate, "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501