# 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

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topMaire, 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- Elliot Benjamin, Franz Lemmermeyer, C. Snyder, Real quadratic fields with abelian $2$-class field tower, J. Number Theory 73 (1998), 182-194 Zbl0919.11073MR1658015
- E. S. Golod, I. R. Šafarevič, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261-272 Zbl0136.02602MR161852
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- René Schoof, Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209-220 Zbl0589.12011MR863524
- W. A. Stein, Sage Mathematics Software (Version 5.11), (Y2013)
- 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

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