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

$K_0(F)\cong 𝐙$

for any field F. Next,

$K_1(F)=F^{\times },$

the multiplicative group of F. 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: for $n\ge 1$,

$K_n({𝔽}_q)={\pi }_n(BGL({𝔽}_q{)}^{+})\simeq \{\begin{array}{cc}\mathbb{Z}/(q^i-1),& \text{if }n=2i-1\\ 0,& \text{if }n\text{ is even}\end{array}$

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 Template:Harvtxt.

Local and global fields

Template:Harvtxt 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

Template:Harvtxt 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

References

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