---
tag: 2YXDQ7Q445
kind: concept
name: Milnor K-theory
msc: [19D45, 12G05]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_K-theory
same_as:
  wikidata: Q6860246
  wikipedia: Milnor K-theory
---

## Summary

Milnor K-theory attaches to a field $K$ a sequence of abelian groups $K_n^M(K)$, see the [[definition|HEVWQ6B77W]].
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|5Q5VN56QH4]] identifies its reduction modulo $\ell$ with [[Galois cohomology|9MPCZ3EAJM]].
It also connects with quadratic forms through the Milnor conjecture.

## Explanations

### L1 human

Start with the nonzero elements of a field under multiplication.
Form "products" of $n$ of them, written $\set{a_1, \ldots, a_n}$, which behave like products in each slot.
Add one extra rule: a symbol vanishes when two of its entries add up to $1$, as in $\set{2, -1} = 0$ over $\Q$.
The groups that result are simple to define, yet they carry deep arithmetic information about the field.