# On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields

Acta Arithmetica (1995)

- Volume: 73, Issue: 1, page 59-65
- ISSN: 0065-1036

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topP. E. Conner, and J. Hurrelbrink. "On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields." Acta Arithmetica 73.1 (1995): 59-65. <http://eudml.org/doc/206810>.

@article{P1995,

abstract = {A large number of papers have contributed to determining the structure of the tame kernel $K₂_F$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K₂_F$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]).
We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms, when the 2-Sylow subgroup of the tame kernel of F is elementary abelian.
This makes determining exactly when the 4-rank of $K₂_F$ is zero, computationally even more accessible. For arbitrary algebraic number fields F with 4-rank of $K₂_F$ equal to zero, it has been pointed out that the Leopoldt conjecture for the prime 2 is valid for F, compare [6].
We consider this paper to be an addendum to the Acta Arithmetica publications [7], [8]. It grew out of our circulated 1989 notes [3].
Acknowledgements. We gratefully acknowledge fruitful long-term communications on this topic with Jerzy Browkin.},

author = {P. E. Conner, J. Hurrelbrink},

journal = {Acta Arithmetica},

keywords = {quadratic number fields; positive definite binary quadratic forms; 2-Sylow subgroup; elementary abelian},

language = {eng},

number = {1},

pages = {59-65},

title = {On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields},

url = {http://eudml.org/doc/206810},

volume = {73},

year = {1995},

}

TY - JOUR

AU - P. E. Conner

AU - J. Hurrelbrink

TI - On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields

JO - Acta Arithmetica

PY - 1995

VL - 73

IS - 1

SP - 59

EP - 65

AB - A large number of papers have contributed to determining the structure of the tame kernel $K₂_F$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K₂_F$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]).
We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms, when the 2-Sylow subgroup of the tame kernel of F is elementary abelian.
This makes determining exactly when the 4-rank of $K₂_F$ is zero, computationally even more accessible. For arbitrary algebraic number fields F with 4-rank of $K₂_F$ equal to zero, it has been pointed out that the Leopoldt conjecture for the prime 2 is valid for F, compare [6].
We consider this paper to be an addendum to the Acta Arithmetica publications [7], [8]. It grew out of our circulated 1989 notes [3].
Acknowledgements. We gratefully acknowledge fruitful long-term communications on this topic with Jerzy Browkin.

LA - eng

KW - quadratic number fields; positive definite binary quadratic forms; 2-Sylow subgroup; elementary abelian

UR - http://eudml.org/doc/206810

ER -

## References

top- [1] P. Barrucand and H. Cohn, Note on primes of type x² + 32y², class number and residuacity, J. Reine Angew. Math. 238 (1969), 67-70. Zbl0207.36202
- [2] P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Sci., Singapore, 1988.
- [3] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂𝓞 containing no element of order four, circulated notes, 1989.
- [4] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂𝓞, Canad. J. Math. 41 (1989), 932-960. Zbl0705.19006
- [5] J. Hurrelbrink, Circulant graphs and 4-ranks of ideal class groups, J. Math. 46 (1994), 169-183. Zbl0792.05133
- [6] M. Kolster, Remarks on étale K-theory and the Leopoldt conjecture, in: Séminaire de Théorie des Nombres, Paris, 1991-92, Progr. Math. 116, Birkhäuser, 1993, 37-62.
- [7] H. Qin, The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields, Acta Arith. 69 (1995), 153-169. Zbl0826.11055
- [8] H. Qin, The 4-rank of $K\u2082{O}_{F}$ for real quadratic fields, Acta Arith. 72 (1995), 323-333.
- [9] B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen, 1970

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