---
tag: HEVWQ6B77W
kind: definition
name: Milnor K-theory
msc: [19D45, 12G05]
defines: 2YXDQ7Q445
uses: [0QA8CHH8MG, 071DDFVG4J, F0HQXQ0HCD, 05THZKR9AN]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_K-theory
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K^\times$ be its [[group of units|05THZKR9AN]], an [[abelian group|071DDFVG4J]].
Let $n \ge 2$.
Let $T_n$ be the [[tensor product|F0HQXQ0HCD]] over $\Z$ of $n$ copies of $K^\times$.
Let $R_n \subseteq T_n$ be the subgroup generated by all $a_1 \otimes \cdots \otimes a_n$ with $a_i + a_j = 1$ for some $i < j$.
The *$n$-th Milnor K-group* of $K$ is the quotient
$$K_n^M(K) = T_n / R_n.$$
Also $K_0^M(K) = \Z$ and $K_1^M(K) = K^\times$.
The class of $a_1 \otimes \cdots \otimes a_n$ is the *symbol* $\set{a_1, \ldots, a_n}$, and the group law of $K_n^M(K)$ is written additively.

## Remarks

Equivalently, $K_*^M(K) = \bigoplus_{n \ge 0} K_n^M(K)$ is the quotient of the tensor algebra of $K^\times$ over $\Z$ by the two-sided ideal generated by the elements $a \otimes (1 - a)$ with $a \ne 0, 1$.
The relation $\set{a, 1 - a} = 0$ is the Steinberg relation.
Consequences: $\set{a, -a} = 0$ and $\set{a, b} = -\set{b, a}$ for all $a, b \in K^\times$.
Example: in $K_2^M(\Q)$, $\set{2, -1} = 0$ since $2 + (-1) = 1$.
Non-example: the element $2 \otimes (-1)$ of $\Q^\times \otimes_\Z \Q^\times$ is not zero.
Indeed $\Q^\times$ is the product of $\set{\pm 1}$ and the free abelian group on the primes.
So $2 \otimes (-1)$ generates a direct summand $\langle 2 \rangle \otimes \set{\pm 1} \cong \Z/2\Z$, where $\langle 2 \rangle$ is the cyclic group generated by $2$.
Thus the tensor product alone is not $K_2^M(\Q)$.
For a finite field $\F_q$, $K_n^M(\F_q) = 0$ for $n \ge 2$.
By a theorem of Matsumoto, $K_2^M(K)$ equals Quillen's $K_2(K)$.