1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of . 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 , where the primes 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 is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed.
@article{bwmeta1.element.bwnjournal-article-aav80i3p225bwm,
author = {A. Vazzana},
title = {On the 2-primary part of K2 of rings of integers in certain quadratic number fields},
journal = {Acta Arithmetica},
volume = {80},
year = {1997},
pages = {225-235},
zbl = {0868.11054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav80i3p225bwm}
}
A. Vazzana. On the 2-primary part of K₂ of rings of integers in certain quadratic number fields. Acta Arithmetica, Tome 80 (1997) pp. 225-235. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav80i3p225bwm/
[000] [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. | Zbl 0207.36202
[001] [CH1] P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Sci., Singapore, 1988.
[002] [CH2] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂𝓞 containing no element of order four, circulated notes, 1989.
[003] [CH3] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂𝓞, Canad. J. Math. 41 (1989), 932-960. | Zbl 0705.19006
[004] [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. | Zbl 0844.11072
[005] [H] J. Hurrelbrink, Circulant graphs and 4-ranks of ideal class groups, Canad. J. Math. 46 (1994), 169-183. | Zbl 0792.05133