On the compositum of all degree $d$  extensions of a number field

Itamar Gal; Robert Grizzard

doi:10.5802/jtnb.884

On the compositum of all degree

d

extensions of a number field

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.

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 655-672.

Abstract (VO)
Résumé

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 $d \leq 4$ . 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 $d \leq 4$ . 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.

EuDML MR Zbl

DOI : 10.5802/jtnb.884

Classification : 12F10, 11R21, 20B05
Keywords: Number fields, infinite algebraic extensions, Galois theory, permutation groups

Affiliations des auteurs :

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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     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},
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}

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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, 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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