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.

We study the compositum k [d] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [d] contains all subextensions of degree less than d if and only if d4. We prove that for d>2 there is no bound c=c(d) on the degree of elements required to generate finite subextensions of k [d] /k. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d, but that one can take c=d when d is prime. This question was inspired by work of Bombieri and Zannier on heights in similar extensions, and previously considered by Checcoli.

Nous étudions le compositum k [d] de toutes les extensions de degré d d’un corps de nombres k dans une clôture algébrique fixée. Nous démontrons que k [d] contient toutes les sous-extensions de degré inférieur à d si et seulement si d4. Nous montrons que quand d>2, il n’existe pas de majorant c=c(d) sur le degré des éléments nécessaires pour engendrer les sous-extensions finies de k [d] /k. En se restreignant aux sous-extensions galoisiennes, nous montrons qu’un tel majorant n’existe pas sous certaines conditions sur les diviseurs de d, mais que l’on peut prendre c=d quand d est premier. Cette question a été inspirée par les travaux de Bombieri et Zannier sur les hauteurs dans des extensions similaires, et examinés par Checcoli.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.884
Classification: 12F10,  11R21,  20B05
Keywords: Number fields, infinite algebraic extensions, Galois theory, permutation groups
Itamar Gal 1; Robert Grizzard 2

1 Department of Mathematics, University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
2 Department of Mathematics University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/

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