On the 2-primary part of K₂ of rings of integers in certain quadratic number fields

A. Vazzana

Acta Arithmetica (1997)

  • Volume: 80, Issue: 3, page 225-235
  • ISSN: 0065-1036

Abstract

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1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of K ₂ E . For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form ℚ ( √ ( p ₁ . . . p k ) ) , where the primes p i are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of K ₂ E is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed.

How to cite

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A. Vazzana. "On the 2-primary part of K₂ of rings of integers in certain quadratic number fields." Acta Arithmetica 80.3 (1997): 225-235. <http://eudml.org/doc/207039>.

@article{A1997,
abstract = {1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K₂_E$. For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $ℚ(√(p₁ ...p_k))$, where the primes $p_i$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K₂_E$ is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed.},
author = {A. Vazzana},
journal = {Acta Arithmetica},
keywords = {2-primary part; of rings of integers},
language = {eng},
number = {3},
pages = {225-235},
title = {On the 2-primary part of K₂ of rings of integers in certain quadratic number fields},
url = {http://eudml.org/doc/207039},
volume = {80},
year = {1997},
}

TY - JOUR
AU - A. Vazzana
TI - On the 2-primary part of K₂ of rings of integers in certain quadratic number fields
JO - Acta Arithmetica
PY - 1997
VL - 80
IS - 3
SP - 225
EP - 235
AB - 1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K₂_E$. For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $ℚ(√(p₁ ...p_k))$, where the primes $p_i$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K₂_E$ is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed.
LA - eng
KW - 2-primary part; of rings of integers
UR - http://eudml.org/doc/207039
ER -

References

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  1. [BC] 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. [CH1] P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Sci., Singapore, 1988. 
  3. [CH2] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂𝓞 containing no element of order four, circulated notes, 1989. 
  4. [CH3] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂𝓞, Canad. J. Math. 41 (1989), 932-960. Zbl0705.19006
  5. [CH4] P. E. Conner and J. Hurrelbrink, On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields, Acta Arith. 73 (1995), 59-65. Zbl0844.11072
  6. [H] J. Hurrelbrink, Circulant graphs and 4-ranks of ideal class groups, Canad. J. Math. 46 (1994), 169-183. Zbl0792.05133

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