Milnor K-theory

Summary

Milnor K-theory attaches to a field $K$ a sequence of abelian groups $K_n^M(K)$, see the definition. They are generated by symbols $\set{a_1, \ldots, a_n}$ of nonzero elements. The symbols are multilinear and satisfy the Steinberg relation $\set{a, 1 - a} = 0$. In degrees $0$, $1$ and $2$ it agrees with the algebraic K-theory groups $\Z$, $K^\times$ and $K_2(K)$. The norm residue isomorphism theorem identifies its reduction modulo $\ell$ with Galois cohomology. It also connects with quadratic forms through the Milnor conjecture.