Comptes Rendus
no. 1-2
Number Theory
On fields of algebraic numbers with bounded local degrees

Presented by: Christophe Soulé

Sara Checcoli1; Umberto Zannier2
1 Dipartimento di Matematica, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
2 Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 11-14.

It is well known that if a field K⊆Q¯ 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 K⊆Q¯ 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.

Received:
Accepted:
Published online:
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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