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
for 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 formula_3,
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 .
Local and global fields.
 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.
 showed that the torsion in K-theory is insensitive to extensions of algebraically closed fields. This statement is known as Suslin rigidity.