Comptes Rendus
Number Theory
On fields of algebraic numbers with bounded local degrees
[Corps de nombres algébriques à degrés locaux bornés]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 11-14.

It is well known that if a field KQ¯ is contained in the compositum of all extensions of Q of degree at most d, then it has uniformly bounded local degrees. One may ask whether the converse holds. The answer is easily seen to be affirmative if the extension K/Q is abelian, but we provide a counterexample to the general assertion. This is built up from a certain family of pq-groups.

Il est bien connu que si un corps KQ¯ est contenu dans le compositum de toutes les extensions de Q de degré inférieur à d, alors il est à degrés locaux uniformément bornés. On se demande si la réciproque est vraie. On prouve facilement que c'est le cas si l'extension K est abélienne, mais cela n'est pas vrai dans le cas général, comme le montre un contre-exemple construit à partir d'une certaine famille de pq-groupes.

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DOI : 10.1016/j.crma.2010.12.007

Sara Checcoli 1 ; Umberto Zannier 2

1 Dipartimento di Matematica, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
2 Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
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Sara Checcoli; Umberto Zannier. On fields of algebraic numbers with bounded local degrees. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 11-14. doi : 10.1016/j.crma.2010.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.12.007/

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[3] W. Narkiewicz Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin, 1990

[4] J. Neukirch; A. Schmidt; K. Wingberg Cohomology of Number Fields, Grundl. Math. Wiss., vol. 323, Springer, 1999

[5] J.-P. Serre Une « formule de masse » pour les extensions totalement ramifiées de degré donné d'un corps local, C. R. Acad. Sci. Paris, Volume 286 (1978), pp. 1031-1036

[6] I.R. Shafarevich On p-extensions, AMS. Transl., Ser. 2, Volume 4 (1956), pp. 59-72

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  • Linda Frey Small heights in large non-abelian extensions, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, Volume 23 (2022) no. 3, pp. 1357-1393 | DOI:10.2422/2036-2145.201811_018 | Zbl:1507.11048
  • James P. Kelly; Charles L. Samuels Direct limits of adèle rings and their completions, Rocky Mountain Journal of Mathematics, Volume 50 (2020) no. 3, pp. 1021-1043 | DOI:10.1216/rmj.2020.50.1021 | Zbl:1453.11151
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  • Itamar Gal; Robert Grizzard On the compositum of all degree d extensions of a number field, Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 655-672 | DOI:10.5802/jtnb.884 | Zbl:1360.11112
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  • Sara Checcoli Fields of algebraic numbers with bounded local degrees and their properties, Transactions of the American Mathematical Society, Volume 365 (2013) no. 4, pp. 2223-2240 | DOI:10.1090/s0002-9947-2012-05712-6 | Zbl:1281.11098

Cité par 8 documents. Sources : zbMATH

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