On p 2 -Ranks in the Class Field Tower Problem  [ p 2 -rangs et p-tours de Hilbert ]
Annales Mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 57-68.

Les récents progrès sur le problème de la 2-tour de Hilbert des corps de nombres portent sur l’infinitude – en particulier pour les corps quadratiques – quand le groupe des classes a un grand 4-rang. Généralisant à tout nombre premier p, nous utilisons les inégalités de type Golod-Safarevic afin d’analyser la contribution du p 2 -rang du groupe des classes à l’étude de la p-tour de Hilbert. Nous apportons également des résultats partiels en direction de l’infinitude de le 2-tour de Hilbert des corps quadratiques réels lorsque que le 2-rang du groupe des classes vaut 5.

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.

DOI : https://doi.org/10.5802/ambp.342
Classification : 11R29,  11R34,  11R37
Mots clés : Tours de Hilbert des corps de nombres
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     author = {Maire, Christian and McLeman, Cam},
     title = {On $p^2$-Ranks in the Class Field Tower Problem},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {57--68},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {21},
     number = {2},
     year = {2014},
     doi = {10.5802/ambp.342},
     mrnumber = {3322615},
     language = {en},
     url = {www.numdam.org/item/AMBP_2014__21_2_57_0/}
}
Maire, Christian; McLeman, Cam. On $p^2$-Ranks in the Class Field Tower Problem. Annales Mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 57-68. doi : 10.5802/ambp.342. http://www.numdam.org/item/AMBP_2014__21_2_57_0/

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