On the compositum of all degree d extensions of a number field
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 655-672.

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.

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.

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DOI : https://doi.org/10.5802/jtnb.884
Classification : 12F10,  11R21,  20B05
Mots clés : Number fields, infinite algebraic extensions, Galois theory, permutation groups
@article{JTNB_2014__26_3_655_0,
     author = {Itamar Gal and Robert Grizzard},
     title = {On the compositum of all degree $d$  extensions of a number field},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {655--672},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {3},
     year = {2014},
     doi = {10.5802/jtnb.884},
     zbl = {06561053},
     mrnumber = {3320497},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/}
}
Itamar Gal; Robert Grizzard. On the compositum of all degree $d$  extensions of a number field. Journal de Théorie des Nombres de Bordeaux, Tome 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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