K-groups of a field

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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

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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.

Finite fields

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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).

Local and global fields

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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).

Algebraically closed fields

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

See also

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References

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  1. ^ Weibel 2013, Ch. III, Example 1.1.2.
  2. ^ Weibel 2013, Ch. IV, Corollary 1.13.
  • Jardine, J. F. (1993), "The K-theory of finite fields, revisited", K-Theory, 7 (6): 579–595, doi:10.1007/BF00961219, MR 1268594
  • Suslin, Andrei (1983), "On the K-theory of algebraically closed fields", Inventiones Mathematicae, 73 (2): 241–245, doi:10.1007/BF01394024, MR 0714090