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 \mathbf {Z}

for any field F. Next,

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 $n\geq 1$ ,

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.

References

Weibel 2013, Ch. III, Example 1.1.2.
Weibel 2013, Ch. IV, Corollary 1.13.

Jardine, J. F. (1993), "The K-theory of finite fields, revisited", K-Theory, 7 (6): 579–595, doi:10.1007/BF00961219, MR 1268594
Suslin, Andrei (1983), "On the K-theory of algebraically closed fields", Inventiones Mathematicae, 73 (2): 241–245, doi:10.1007/BF01394024, MR 0714090

Weibel, Charles (2005), "Algebraic K-Theory of Rings of Integers in Local and Global Fields", in Friedlander, Eric M.; Grayson, Daniel R. (eds.), Handbook of K-Theory, Springer, pp. 139–190, doi:10.1007/978-3-540-27855-9_5, ISBN 978-3-540-27855-9

Weibel, Charles A. (2013), The K-book, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, ISBN 978-0-8218-9132-2, MR 3076731

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[1] Weibel 2013, Ch. III, Example 1.1.2.

[2] Weibel 2013, Ch. IV, Corollary 1.13.