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