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.
The map sending a finite-dimensional F-vector space to its dimension induces an isomorphism
for any field F. Next,
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.
The K-groups of finite fields are one of the few cases where the K-theory is known completely:[2] for ,
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).
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).
Suslin (1983) showed that the torsion in K-theory is insensitive to extensions of algebraically closed fields. This statement is known as Suslin rigidity.