On p 2 -Ranks in the Class Field Tower Problem

Christian Maire[1]; Cam McLeman[2]

  • [1] Laboratoire de Mathématiques UMR 6623 CNRS - Université de Franche-Comté 16, Route de Gray 25030 Besançon cedex FRANCE
  • [2] Mathematics Department University of Michigan - Flint 303 E. Kearsley St. Flint, MI, 48502 USA

Annales mathématiques Blaise Pascal (2014)

  • Volume: 21, Issue: 2, page 57-68
  • ISSN: 1259-1734

Abstract

top
Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the p 2 -rank of the class group as a quantity of relevance in the p -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups have a 2-rank of 5 must have an infinite 2-class field tower.

How to cite

top

Maire, Christian, and McLeman, Cam. "On $p^2$-Ranks in the Class Field Tower Problem." Annales mathématiques Blaise Pascal 21.2 (2014): 57-68. <http://eudml.org/doc/275447>.

@article{Maire2014,
abstract = {Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^2$-rank of the class group as a quantity of relevance in the $p$-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups have a 2-rank of 5 must have an infinite 2-class field tower.},
affiliation = {Laboratoire de Mathématiques UMR 6623 CNRS - Université de Franche-Comté 16, Route de Gray 25030 Besançon cedex FRANCE; Mathematics Department University of Michigan - Flint 303 E. Kearsley St. Flint, MI, 48502 USA},
author = {Maire, Christian, McLeman, Cam},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Hilbert class field towers},
language = {eng},
month = {7},
number = {2},
pages = {57-68},
publisher = {Annales mathématiques Blaise Pascal},
title = {On $p^2$-Ranks in the Class Field Tower Problem},
url = {http://eudml.org/doc/275447},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Maire, Christian
AU - McLeman, Cam
TI - On $p^2$-Ranks in the Class Field Tower Problem
JO - Annales mathématiques Blaise Pascal
DA - 2014/7//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 2
SP - 57
EP - 68
AB - Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^2$-rank of the class group as a quantity of relevance in the $p$-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups have a 2-rank of 5 must have an infinite 2-class field tower.
LA - eng
KW - Hilbert class field towers
UR - http://eudml.org/doc/275447
ER -

References

top
  1. Elliot Benjamin, Franz Lemmermeyer, C. Snyder, Real quadratic fields with abelian 2 -class field tower, J. Number Theory 73 (1998), 182-194 Zbl0919.11073MR1658015
  2. E. S. Golod, I. R. Šafarevič, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261-272 Zbl0136.02602MR161852
  3. Christian Maire, Un raffinement du théorème de Golod-Safarevic, Nagoya Math. J. 150 (1998), 1-11 Zbl0923.11158MR1633138
  4. René Schoof, Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209-220 Zbl0589.12011MR863524
  5. W. A. Stein, Sage Mathematics Software (Version 5.11), (Y2013) 
  6. B. B. Venkov, H. Koh, The p -tower of class fields for an imaginary quadratic field, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 46 (1974), 5-13, 140 Zbl0335.12022MR382235

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.