K-groups of a field

In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen.

Low degrees

The map sending a finite-dimensional F-vector space to its dimension induces an isomorphism

${\displaystyle K_{0}(F)\cong \mathbf {Z} }$

for any field F. Next,

${\displaystyle K_{1}(F)=F^{\times },}$

the multiplicative group of F.^[1] The second K-group of a field is described in terms of generators and relations by Matsumoto's theorem.

Finite fields

The K-groups of finite fields are one of the few cases where the K-theory is known completely:^[2] for ${\displaystyle n\geq 1}$,

${\displaystyle K_{n}(\mathbb {F} _{q})=\pi _{n}(BGL(\mathbb {F} _{q})^{+})\simeq {\begin{cases}\mathbb {Z} /{(q^{i}-1)},&{\text{if }}n=2i-1\\0,&{\text{if }}n{\text{ is even}}\end{cases}}}$

For n=2, this can be seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture. A different proof was given by Jardine (1993).

Local and global fields

Weibel (2005) surveys the computations of K-theory of global fields (such as number fields and function fields), as well as local fields (such as p-adic numbers).

Algebraically closed fields

Suslin (1983) showed that the torsion in K-theory is insensitive to extensions of algebraically closed fields. This statement is known as Suslin rigidity.

See also

• divisor class group