The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields

Hourong Qin

Acta Arithmetica (1995)

  • Volume: 69, Issue: 2, page 153-169
  • ISSN: 0065-1036

Abstract

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1. Introduction. Let F be a number field and O F the ring of its integers. Many results are known about the group K ₂ O F , the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of K ₂ O F . As compared with real quadratic fields, the 2-Sylow subgroups of K ₂ O F for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of K ₂ O F for imaginary quadratic fields F. In our Ph.D. thesis (see [11]), we develop a method to determine the structure of the 2-Sylow subgroups of K ₂ O F for real quadratic fields F. The present paper is motivated by some ideas in the above thesis.

How to cite

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Hourong Qin. "The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields." Acta Arithmetica 69.2 (1995): 153-169. <http://eudml.org/doc/206678>.

@article{HourongQin1995,
abstract = {1. Introduction. Let F be a number field and $O_F$ the ring of its integers. Many results are known about the group $K₂O_F$, the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of $K₂O_F$. As compared with real quadratic fields, the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F. In our Ph.D. thesis (see [11]), we develop a method to determine the structure of the 2-Sylow subgroups of $K₂O_F$ for real quadratic fields F. The present paper is motivated by some ideas in the above thesis.},
author = {Hourong Qin},
journal = {Acta Arithmetica},
keywords = {tame kernel; imaginary quadratic field; Tate kernel; 4-rank},
language = {eng},
number = {2},
pages = {153-169},
title = {The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields},
url = {http://eudml.org/doc/206678},
volume = {69},
year = {1995},
}

TY - JOUR
AU - Hourong Qin
TI - The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields
JO - Acta Arithmetica
PY - 1995
VL - 69
IS - 2
SP - 153
EP - 169
AB - 1. Introduction. Let F be a number field and $O_F$ the ring of its integers. Many results are known about the group $K₂O_F$, the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of $K₂O_F$. As compared with real quadratic fields, the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F. In our Ph.D. thesis (see [11]), we develop a method to determine the structure of the 2-Sylow subgroups of $K₂O_F$ for real quadratic fields F. The present paper is motivated by some ideas in the above thesis.
LA - eng
KW - tame kernel; imaginary quadratic field; Tate kernel; 4-rank
UR - http://eudml.org/doc/206678
ER -

References

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  1. [1] B. Brauckmann, The 2-Sylow subgroup of the tame kernel of number fields, Canad. J. Math. 43 (1991), 215-264. Zbl0729.11061
  2. [2] J. Browkin and A. Schinzel, On Sylow 2-subgroups of K ₂ O F for quadratic fields F, J. Reine Angew. Math. 331 (1982), 104-113. 
  3. [3] J. W. S. Cassels, Rational Quadratic Forms, Academic Press, London, 1978. 
  4. [4] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂(O) containing no elements of order four, preprint. Zbl0844.11072
  5. [5] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂(O), Canad. J. Math. 41 (1989), 932-960. Zbl0705.19006
  6. [6] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York, 1982. Zbl0482.10001
  7. [7] M. Kolster, The structure of the 2-Sylow subgroup of K₂(O), I , Comment. Math. Helv. 61 (1986), 576-588. 
  8. [8] J. Milnor, Introduction to Algebraic K-theory, Ann. of Math. Stud. 72, Princeton University Press, 1971. Zbl0237.18005
  9. [9] J. Neukirch, Class Field Theory, Springer, Berlin, 1986. Zbl0587.12001
  10. [10] O. T. O'Meara, Introduction to Quadratic Forms, Springer, Berlin, 1963. 
  11. [11] H. Qin, K₂ and algebraic number theory, Ph.D. Thesis, Nanjing University, 1992. 
  12. [12] H. Qin, The 2-Sylow subgroups of K ₂ O F for real quadratic fields F, Science in China Ser. A 23 (12) (1993), 1254-1263. 
  13. [13] J. Tate, Relations between K₂ and Galois cohomology, Invent. Math. 36 (1976), 257-274. Zbl0359.12011

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