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''.
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 $n \ge 1$,
:$K_n(\mathbb_q) = \pi_n(BGL(\mathbb_q)^+) \simeq \begin \mathbb/, & \textn = 2i - 1 \\ 0, & \textn\text \end$
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.

See also

*divisor class group

References

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Algebraic geometry

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