# Milnor K-theory

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by Milnor (1970).

## Definition

The calculation of K2 of a field F led Milnor to the following ad hoc definition of "higher" K-groups by

$K^M_*(F) := T^*F^\times/(a\otimes (1-a)), \,$

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

$a\otimes(1-a) \,$

for a ≠ 0, 1. For n = 0,1,2 these coincide with Quillen's K-groups of a field, but for n ≧ 3 they differ in general. We define the symbol $\{a_1,\ldots,a_n\}$ as the image of $a_1 \otimes \cdots \otimes a_n$: the case n=2 is a Steinberg symbol.[1]

The tensor product on the tensor algebra induces a product $K_m \times K_n \rightarrow K_{m+n}$ making $K^M_*(F)$ a graded ring which is graded-commutative.[2]

## Examples

For example, we have $K^M_n(\mathbb{F}_q) = 0$ for n ≧ 2; $K^M_2(\mathbb{C})$ is an uncountable uniquely divisible group; $K^M_2(\mathbb{R})$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; $K^M_2(\mathbb{Q}_p)$ is the direct sum of the multiplicative group of $\mathbb{F}_p$ and an uncountable uniquely divisible group; $K^M_2(\mathbb{Q})$ is the direct sum of the cyclic group of order 2 and cyclic groups of order $p-1$ for all odd prime $p$.

## Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing $K^M_1$ 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

$\{a_1,\ldots,a_n\} \mapsto \langle \langle a_1, a_2, ... , a_n \rangle \rangle = \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle \ ,$

where the image is a Pfister form of dimension 2n.[1] The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In.[3] The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.[1]

## References

1. ^ a b c Lam (2005) p.366
2. ^ Gille & Szamuely (2006) p.184
3. ^ Lam (2005) p.316