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 isomorphismfor any field F. Next,
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,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.